Resonance Rank Booster For Physics

 JEE (ADVANCED) - RRB 

TOPIC

STRAIGHT LINE

1

SECTION - I : STRAIGHT OBJECTIVE TYPE 1.1

Given the family of lines, a (3x + 4y + 6) + b (x + y + 2) = 0 . The line of the family situated at the greatest distance from the point P (2, 3) has equation (A) 4x + 3y + 8 = 0 (B) 5x + 3y + 10 = 0 (C) 15x + 8y + 30 = 0 (D) none

1.2

If two vertices of a triangle are (–2 ,3) and (5 , –1) , orthocentre lies at the origin and centroid on the line x + y = 7 , then the third vertex is (A) (7, 4) (B) (8, 14) (C) (12, 21) (D) None of these

1.3

OPQR is a square and M, N are the mid points of the sides PQ and QR respectively. If the ratio of the areas of the square and the triangle OMN is λ : 6, then (A) 2

1.4

(B) 4

 is equal to 4 (C) 12

(D) 16

3 | x |  2 at a distance of 5 units from their point of intersection. The co-ordinates of the foot of the perpendicular from P on the bisector of the angle between them are :-

P is point on either of the two lines y –

 45 3   45 3      0, (A)  0, or   depending on which the points p is taken  2 2      45 3    (B)  0,  2    45 3    (C)  0,  2   1.5

5 5 3    (D)  2 , 2   

A pair of perpendicular straight lines drawn through the origin form an isosceles triangle with line 2x + 3y = 6, then area of the triangle so formed is (A)

36 13

(B)

12 17

(C)

13 5

(D)

17 13

1.6

Let ABC be a triangle. Let A be the point (1, 2), y = x is the perpendicular bisector of AB and x – 2y + 1 = 0 is the angle bisector of ∠C. If equation of BC is given by ax + by – 5 = 0, then the value of a + b is (A) 1 (B) 2 (C) 3 (D) 4

1.7

The distance of the line 2x – 3y = 4 from the point (1, 1) in the direction of the line x + y = 1 is (A)

2

(B) 5 2

(C)

1 2

(D) None of these

1.8

If A(2, 1), B(8, 1), C(4, 3) and D(6, 6), then the area of the quadrilateral ABDC is (A) 14 units (B) 7 units (C) 28 units (D) none of these

1.9

The image of P(a, b) in the line y = – x is Q and the image of Q in the line y = x is R, then the mid-point of PR is (A) (a + b, b + a)

ab b a ,  (B)  2   2

(C) (a – b, b – a)

(D) (0, 0) 1

 JEE (ADVANCED) - RRB 

1.10

If in triangle ABC , A ≡ (1, 10) , circumcentre ≡

 13 , 23 

and orthocentre ≡

113 , 43  then the co-

ordinates of mid-point of side opposite to A is 11   (A) 1,   3 

(B) (1, 5)

(C) (1, − 3)

(D) (1, 6)

1.11

A rectangle ABCD has its side AB parallel to the line y = x and vertices A, B and D lie on y = 1, x = 2 and x = – 2 respectively. Locus of vertex C is (A) x – y = 5 (B) x = 5 (C) x + y = 5 (D) y = 5

1.12

The sides of a rectangle are x = 0, y = 0, x = 4 and y = 3. The equation of the straight line having

1 that divides the rectangle in to two equal halves, is 2 (A) 2x + y = 1 (B) 2x = y + 1 (C) 2y = x + 1 slope

1.13

Consider the point A(3, 4), B(7, 13). If ‘P’ be a point on the line y = x such that PA + PB is minimum, then coordinates of ‘P’ is  13 13   (A)  ,  7 7 

1.14

 23 23   (B)  ,  7 7 

 31 31   (C)  ,  7 7

mn m n = + . Find the locus of P which is a straight line OP OR OS

passing through the point of intersection of L1 and L2. (A) cn (ax + by – 1) + m(y – c) = 0 (C) cn (ax + by – 1) + (y – c) = 0

(B) n (ax + by – 1) + m(y – c) = 0 (D) n (ax + by – 1) + (y – c) = 0

Chords of the curve 4x2 + y2 – x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose co-ordinates are :  1 4 (A)  – ,   5 5

1.16

 33 33   (D)  ,  7 7 

A variable line is drawn through O to cut two fixed straight lines L1 and L 2 in R and S. A point P is chosen on the variable line such that

1.15

(D) 2y + x = 1

1 –4  (B)  , 5 5 

1 4 (C)  ,  5 5

 –1 – 4 ,  (D)  5   5

Which of the following statement(s) is/are correct ? S1 : The new co-ordinates of a point (4, 5), when the origin is shifted to the point (1, –2) are (3, 7) . S2 : Locus of a point whose distance from (a, 0) is equal to its distance from y-axis, is y2 – 2ax + a2 = 0. S3 : If the point (a, a) is placed in between the lines |x + y| = 4, then |a| = 2. S4 : If A(2, 2) , B(–4, –4), C(5, –8) are vertices of any triangle, then the length of median pass through C is

85 . (A) TFFT 1.17

(B) TFFF

(C) TTFT

(D) TTTF

Which of the following statement(s) is/are correct ? S1 : The lines x + (loga b) y + (loga c) = 0, (logba) x + y + (logb c) = 0 and (logca)x + (logc b) y+1 = 0 are concurrent S2 : Equation of a straight line passing through the origin and making with positive x − axis an angle twice the size of the angle made by the line y = 0.2 x with the positive x − axis, is y = 0.4 x

27 sq. units. 4 S4 : Lines through the origin and perpendicular to the lines xy – 3 y2 + y – 2 x + 10 = 0 is 3 y2 + x y = 0 S3 : Area of the triangle formed by the lines y 2 – 9xy + 18x2 = 0 and y = 9 is

(A) FTTF

(B) TFTF

(C) TTFF

(D) FFTT

2

 JEE (ADVANCED) - RRB 

1.18

Consider the following statements : S1 : The image of the point (2, 1) with respect to the line x + 1 = 0 is (–2, 1). S2 : If (, m) is a point on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10, then

m 2

is a prime number. S3 : Orthocentre of the triangle with vertices (10, 20), (22, 25) and (10, 25) is (10, 25). S4 : The line y = mx bisect the angle between the lines ax2 – 2hxy + by2 = 0 if h(1 – m2) + m (a – b) = 0 State, in order, whether S1, S2, S3, S4 are true or false (A) FTTF

(B) FFTT

(C) TTFF

(D) FTTT

SECTION - II : MULTIPLE CORRECT ANSWER TYPE 1.19

The sides of a triangle are the straight lines x + y = 1 ; 7y = x and 3 y + x = 0. Then which of the following is an interior point of the triangle ? (A) circumcentre (B) centroid (C) incentre (D) orthocentre

1.20

If one diagonal of a square is the portion of the line

x y  = 1 intercepted by the axes, then the a b

extremities of the other diagonal of the square are

 a  b a  b ,   2 2 

 a  b a  b ,   2 2 

(A)  1.21

1.22

4 

(D) 

3 , 3 3



(B)

4 

3 , 3 3



(C)

3 

3 , 4 3



(D)

3 

3 , 4 3



The points A (0 , 0), B (cos α , sin α) and C (cos β, sin β) are the vertices of a right angled triangle if (A) sin

1.24

 a  b ba ,   2 2 

(C) 

Two straight lines u = 0 and v = 0 passes through the origin and angle between them is tan −1 (7/9). If the ratio of the slope of v = 0 and u = 0 is 9/2, then their equations are (A) y = 3x & 3y = 2x (B) 2y = 3x & 3y = x (C) y + 3x = 0 & 3y + 2x = 0 (D) 2y + 3x = 0 & 3y + x = 0 A and B are two fixed points whose co-ordinates are (3, 2) and (5, 4) respectively. The co-ordinates of a point P if ABP is an equilateral triangle, are (A)

1.23

 a  b ba ,   2 2 

(B) 

1   = 2 2

(B) cos

1   1 =− (C) cos = 2 2 2 2

(D) sin

1  =− 2 2

If x – 2y + 4 = 0 and 2x + y – 5 = 0 are the sides of an isosceles triangle having area 10 sq. units, then equation of third side is (A) x + 3y + 10 = 0 (B) 3x – y + 9 = 0 (C) x + 3y – 19 = 0 (D) 3x – y – 11 = 0

SECTION - III : ASSERTION AND REASON TYPE ,

1.25

STATEMENT - 1 : The internal angle bisector of angle C of a triangle ABC with sides AB, AC and BC are y = 0 , 3x + 2y = 0 and 2x + 3y + 6 = 0 respectively, is 5x + 5y + 6 = 0. STATEMENT -2 : Image of point A with respect to 5x + 5y + 6 = 0 lies on side BC of the triangle. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

1.26

STATEMENT -1 : Let the lines 2x + 3y + 19 = 0 and 9x + 6y – 17 = 0 cut the x axis in A, B and y axis in C,D respectively, then points A,B, C, D are concyclic. STATEMENT - 2 : Since OA . OB = OC . OD, where O is origin therefore A, B, C, D points are concyclic (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 3

 JEE (ADVANCED) - RRB 

1.27

STATEMENT-1 : Perpendicular from point A (1, 1) to the line joining the points B (c cos α, c sin α) and C (c cos β, c sin β) bisects BC for all values of α and β. and STATEMENT-2 : Perpendicular drawn from the vertex to the base of an isosceles triangle bisects the base. (A) (B) (C) (D)

STATEMENT-1 STATEMENT-1 STATEMENT-1 STATEMENT-1 STATEMENT-1 STATEMENT-1

is True, STATEMENT-2 is True ; STATEMENT-2 is a correct explanation for is True, STATEMENT-2 is True ; STATEMENT-2 is NOT a correct explanation for is True, STATEMENT-2 is False is False, STATEMENT-2 is True

1.28

STATEMENT -1 : Let the vertices of a ΔABC are A(–5, –2), B(7, 6) and C(5, –4) , then co-ordinates of circumcentre is (1, 2). STATEMENT -2 : In a right angle triangle, mid-point of hypotenuous is the circumcentre of the triangle. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

1.29

STATEMENT -1 : If – 2h = a + b, then one line of the pair of lines ax2 + 2hxy + by2 = 0 bisects the angle between co-ordinate axes in positive quadrant. STATEMENT -2 : If ax + y(2h + a) = 0 is a factor of ax2 + 2hxy + by2 = 0, then b + 2h + a = 0. ((A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

1.30

STATEMENT -1 : Two of the straight lines represented by the equation ax3 + bx2 y + cxy2 + dy3 = 0 will be right angled if a2 + ac + bc + d2 = 0 . STATEMENT -2 : Product of the slopes of two perpendicular lines is – 1. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

SECTION - IV : COMPREHENSION TYPE Read the following comprehensions carefully and answer the questions. Comprehension # 1 Let P(x1, y1) be a point not lying on the line  : ax + by + c = 0. Let L be a point on line  such that PL is perpendicular to the line  . Let Q(x, y) be a point on the line passing through P and L. Let absolute distance between P and Q is n times (n ∈ R+) the absolute distance between P and L. If L and Q lie on the same side of P, then coordinates of Q are given by the formula

ax1  by1  c x  x1 y  y1 = =–n and if L and Q lie on the opposite sides of P, a b a2  b 2

then the coordinates of Q are given by the formula

ax1  by1  c x  x1 y  y1 = = n a b a2  b 2

1.31

Let (2, 3) be the point P and 3x – 4y + 1 = 0 be the straight line , if the sum of the coordinates of a point Q lying on PL, where L and Q lie on the same side of P and n = 15 is α , then α = (A) 0 (B) 1 (C) 2 (D) 3

1.32

Let (1, 1) be the point P and – 5x + 12y + 6 = 0 be the straight line , if the sum of the coordinates of a point Q lying on PL, where L and Q are on opposite sides of P and n = 13α is β, then β = (α is as obtained in the above question) (A) –9 (B) 25 (C) 12 (D) 16 4

 JEE (ADVANCED) - RRB 

1.33

Let (2, –1) be the point P and x – y + 1 = 0 be the straight line , if a point Q lies on PL where L and Q are on the same side of P for which n = β, then the coordinates of the image Q′ of the point Q in the line  are (β is as obtained in the above question) (A) (14, 28) (B) (30, –29) (C) (26, –27) (D) (–26, 27)

Comprehension # 2 Let us consider the situation when axes are inclined at an angle ‘ω’. If coordinates of a point P are (x1, y1) then PN = x1, PM = y1. Where PM is parallel to y-axis and PN is parallel x-axis. Now RQ = y – y1, PQ = x – x1 From ΔPQR, we have

PQ RQ = sin(  ) sin  ∴ Equation of straight line through P and makes an angle θ with x-axis is y – y1 =

sin  (x – x1) sin(  )

written in the form of y – y1 = m(x – x1) where m = ∴

sin  . (m is called of slope of line) sin(  )

 m sin    Angle of inclination of line with x-axis is given by tan θ =   1  m cos  

Read the above comprehension and answer the following questions.

1.34

The axes being inclined at an angle of 60°, then the inclination of the straight line y = 2x + 5 with the axis of x is

 3  (B) tan–1   2  

(A) 30°

1.35

(D) 60°

The axes being inclined at an angle of 60°, then angle between the two straight lines y = 2x + 5 and 2y + x + 7 = 0 is 5 (B) tan–1   3

(A) 90°

1.36

(C) tan–12

 3  (C) tan–1   2  

 5   (D) tan–1    3

The axes being inclined at an angle of 30°, then equation of straight line which makes an angle of 60º with the positive direction of x-axis and x-intercept equal to 2, is (A) y –

3x=0

(B)

3y=x

(C) y +

3x=2 3

(D) y + 2x = 0

Comprehension # 3 2  2 A(1, 3) and C   ,   are the vertices of a triangle ABC and the equation of the angle bisector of  5 5

∠ABC is x + y = 2. Answer the following questions 1.37

Equation of side BC is (A) 7x + 3y – 4 = 0 (B) 7x + 3y + 4 = 0

(C) 7x – 3y + 4 = 0

(D) 7x – 3y – 4 = 0

5

 JEE (ADVANCED) - RRB 

1.38

1.39

Coordinates of vertex B are  3 17   17 3  (A)  , (B)  ,   10 10    10 10  Equation of side AB is (A) 3x + 7y = 24 (B) 3x + 7y + 24 = 0

 5 9 (C)   ,   2 2

(D) (1, 1)

(C) 13x + 7y + 8 = 0

(D) 13x – 7y + 8 = 0

SECTION - V : MATRIX - MATCH TYPE 1.40

1.41

Column - Ι

Column - ΙΙ

(A)

Two vertices of a triangle are (5, – 1) and (– 2, 3). If orthocentre is the origin, then coordinates of the third vertex are

(p)

(– 4, – 7)

(B)

A point on the line x + y = 4 which lies at a unit distance from the line 4x + 3y = 10, is

(q)

(– 7, 11)

(C)

Orthocentre of the triangle made by the lines x + y – 1 = 0, x – y + 3 = 0, 2x + y = 7 is

(r)

(1, –2)

(D)

If a, b, c are in A.P., then lines ax + by = c are concurrent at

(s) (t)

(–1, 2) (4, –7)

Match the following : Column – Ι

1.42

Column – ΙΙ

(A)

Lines x – 2y – 6 = 0, 3x + y – 4 = 0 and λx + 4y + λ2 = 0 are concurrent , then value of λ is

(p)

2

(B)

The points (λ + 1, 1), (2λ + 1, 3) and (2λ + 2, 2λ) are collinear, then the value of λ is

(q)

4

(C)

If line x + y – 1– λ = 0, passing through the intersection of x – y + 1 = 0 and 3x + y – 5 = 0 is perpendicular to one of them, then the vlaue of λ is

(r)

– 1/2

(D)

If line y – x – 1 + λ = 0 is equally inclined to axes and equidistant from the points (1, –2) and (3, 4), then λ is

(s)

–4

(t)

3

Match the following Column - Ι

Column - ΙΙ

(A) The number of integral values of ‘a’ for which the point P(a, a2) lies completely inside the triangle formed by the lines x = 0, y = 0 and x + 2y = 3

(p)

1

(B) Triangle ABC with AB = 13, BC = 5 and AC = 12 slides on the coordinate axis with A and B on the positive x-axis and positive y-axis respectively, the locus of vertex C is a line 12x – ky = 0, then the value of k is

(q)

4

(C) The reflection of the point (t – 1, 2t + 2) in a line is (2t + 1, t), then the line has slope equals to

(r)

3

(D) In a triangle ABC the bisector of angles B and C lie along the

(s)

5

(t)

0

lines x = y and y = 0. If A is (1, 2) then 10 d(A,BC) where d(A, BC) represents distance of point A from side BC

6

 JEE (ADVANCED) - RRB 

1.43

Match the column Column - I

Column - I

(A)

Area of the region enclosed by 2|x| + 3|y| ≤ 6 is

(p)

12

(B)

The extrimities of the base of an isosceles triangle ABC are the points A(2, 0) and B(0, 1).

(q)

4

(r)

5

(s)

3

(t)

2

If the equation of the side AC is x = 2 and ‘m’ be the slope of side BC, then ‘4m’ equals to (C)

Area of ΔABC is 20 sq. units where points A, B and C are (4, 6), (10, 14) and (x, y) respectively. If AC is perpendicular to BC, then number of positions of C is

(D)

In a ΔABC co-ordinates of orthocentre, centroid and vertex A are respectively (2, 2), (2, 1) and (0, 2). Then x-coordinate of vertex B is

SECTION - VI : INTEGER TYPE 1.44.

The vertices B and C of a triangle ABC lie on the lines 3y = 4x and y = 0 respectively and the side BC 2 2 passes through the point  3 , 3  . If ABOC is a rhombus, O being the origin. If co-ordinates of vertex A  

is (α, β), then find the value of

1.45

5 (α + β). 2

The equations of two adjacent sides of a rhombus formed in first quadrant are represented by 7x2 – 8xy + y2 = 0, then slope of its longer diagonal is :

7

 JEE (ADVANCED) - RRB 

TOPIC

CIRCLE

2

SECTION - I : STRAIGHT OBJECTIVE TYPE 2.1

If r1 and r2 are the radii of smallest and largest circles which passes through (5, 6) and touches the circle (x – 2)2 + y2 = 4, then r1 r2 is

4 41

(A) 2.2

(B)

41 4

(C)

5 41

(D)

41 6

Minimum radius of circle which is orthogonal with both the circles x 2 + y2 – 12x + 35 = 0 and x2 + y2 + 4x +3 = 0 is (A) 4

(B) 3

(C) 15

(D) 1

2.3

S(x, y) = 0 represents a circle. The equation S(x, 2) = 0 gives two identical solutions x = 1 and the equation S(1, y) = 0 gives two distinct solutions y = 0, 2. Find the equation of the circle. (A) x2 + y2 + 2x – 2y + 1 = 0 (B) x2 + y2 – 2x + 2y + 1 = 0 2 2 (D) x2 + y2 – 2x – 2y + 1 = 0 (C) x + y – 2x – 2y – 1 = 0

2.4

From a point R(5, 8) two tangents RP and RQ are drawn to a given circle S = 0 whose radius is 5. If circumcentre of the triangle PQR is (2, 3), then the equation of circle S = 0 is (A) x2 + y2 + 2x + 4y – 20 = 0 (B) x2 + y2 + x + 2y – 10 = 0 2 2 (C) x + y – x – 2y – 20 = 0 (D) x2 + y2 – 4x – 6y – 12 = 0

2.5

Consider a family of circles passing through two fixed points A (3,7) & B(6,5). Find the point of concurrency of the chords in which the circle x2 + y2 – 4x – 6y – 3 = 0 cuts the members of the family :  11 3  ,  (A)   17 7 

 23  (B)  2 ,  3  

(C) (–4 , 3)

(D) chords are not concurrent

2.6

If the tangents are drawn from any point on the line x + y = 3 to the circle x2 + y2 = 9, then the chord of contact passes through the point (A) (3, 5) (B) (3, 3) (C) (5, 3) (D) none of these

2.7

The triangle PQR is inscribed in the circle x2 + y2 = 25 such that P lies on the major arc QR. If Q and R have coordinates (3, 4) and (–4, 3) respectively, then ∠ QPR is equal to (A)

2.8

2.9

 2

(B)

(C)

 4

(D)

 6

Equation of chord of the circle x2 + y2 – 3x – 4y – 4 = 0, which passes through the origin such that origin divides it in the ratio 4 : 1, is (A) x = 0 (B) 24x + 7y = 0 (C) 7x + 24y = 0 (D) 7x – 24y = 0 If the radius of the circumcircle of the triangle TPQ, where PQ is chord of contact corresponding to point T with respect to circle x2 + y2 – 2x + 4y – 11 = 0, is 6 units, then minimum distance of T from the director circle of the given circle is: (A) 6

2.10

 3

(B) 12

(C) 6 2

(D) 12 – 4 2

P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the co−ordinate axes cut at right angles, then (A) a2

− 6ab + b2 = 0

(B) a2 + 2ab

− b2 = 0

(C) a2

− 4ab + b2 = 0

(D) a2

− 8ab + b2 = 0 8

 JEE (ADVANCED) - RRB 

2.11

The exhaustive range of values of ‘a’ such that the angle between the pair of tangents drawn from (a, a) to the   circle x2 + y2 – 2x – 2y – 6 = 0 lies in the range  ,   , is 3  (A) (1, ∞) (B) (–5, –3) ∪ (3, 5)

(C) (– ∞ , – 2 2 ) ∪ (2 2 , ∞)

(D) (–3, –1) ∪ (3, 5)

2.12

In triangle ABC equation of side BC is x – y = 0 circumcentre and orthocentre of the triangle are (2,3) and (5,8) respectively. Equation of circumcircle of the triangle is (B) x2 + y2 – 4x – 6y – 27 = 0 (A) x2 + y2 – 4x + 6y – 27 = 0 2 2 (C) x + y + 4x + 6y – 27 = 0 (D) x2 + y2 + 4x – 6y – 27 = 0

2.13.

A circle touches the lines y 

x

, y = x 3 and has unit radius. If the centre of this circle lies in the first 3 quadrant, then one possible equation of this circle is (A) x2 + y2 – 2x ( 3 + 1) – 2y ( 3 +1) + 8 + 4 3 = 0 (B) x2 + y2 – 2x ( 3 + 1) – 2y ( 3 +1) + 5 + 4 3 = 0 (C) x2 + y2 – 2x ( 3 + 1) – 2y ( 3 +1) + 7 + 4 3 = 0 (D)

x2 + y2 – 2x ( 3 + 1) – 2y ( 3 +1) + 6 + 4 3 = 0

2.14

Equation of the straight line, which meets the circle x2 + y2 = 100 in two points, each point at a distance of 4 unit from the point (8,6), is (A) 4x + 3y – 50 = 0 (B) 4x + 3y – 100 = 0 (C) 4x + 3y – 46 = 0 (D) none of these

2.15

A light ray gets reflected from the line x = – 2. If the reflected ray touches the circle x2 + y2 = 4 and point of incident is (–2, –4), then equation of incident ray is (A) 3x + 4y + 22 = 0 (B) 4x + 3y + 20 = 0 (C) x + 2y + 10 = 0 (D) x + y + 6 = 0

2.16

S1 :

The locus of the centre of a circle which cuts a given circle orthogonally and also touches a given straight line is a parabola.

S2 :

Two circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touches iff

S3 : S4 :

ab a  b2 2

(A) TFTF S1 :

S3 :

2

=

1

. (B) TTFF

(C) TFTT

(D) FFTT

If the length of tangent drawn from an external point P to the circle of radius r is  , then area of triangle formed by pair of tangents and its chord of contact is

S2 :

2

+

1

. a b c2 The two circles which passes through (0, a) and (0, –a) and touch the straight line y = mx + c, will cut orthogonally if c2 = a 2 (2 + m2). The length of the common chord of the circles (x – a)2 + y2 = a2 and x2 + (y – b)2 = b2 is

2.17

1

r 3

. r 2  2 If the points where the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 meet the co-ordinate axes are concyclic, then a1c1 = a2c2 A circle is inscribed in an equilateral triangle of side a, the area of any square inscribed in the a2 8 The equation of the circle with origin as centre passing the vertices of an equilateral triangle whose median is of length 3a is x2 + y2 = 4a2

circle is S4 :

(A) F F T T

(B) T T T F

(C) T F F T

(D) T T T T 9

 JEE (ADVANCED) - RRB 

2.18

2

2

S1 : If the point (0, g) lies inside the circle x + y + 2gx + c = 0, then c cannot be positive. S2 : Length of tangent from origin to the circle 4x2 + 4y2 + 8x + 8y + 1 = 0 is

1 . 2

S3 : The equation 2x2 + 3y2 – 8x – 18y + 35 = k represents a point if k = 0. S4 : The point (λ, 1 + λ) lies inside the circle x2 + y2 = 1 if λ = – (A) F F T T

(B) T T T F

1 . 2

(C) T F F F

(D) T T T T

SECTION - II : MULTIPLE CORRECT ANSWER TYPE 2.19

Consider the circle x2 + y2 – 10x – 6y + 30 = 0. Let O be the centre of the circle and tangent at A(7, 3) and B(5, 1) meet at C. Let S = 0 represents family of circles passing through A and B, then (A) area of quadrilateral OACB = 4 (B) the radical axis for the family of circles S = 0 is x + y = 10 (C) the smallest possible circle of the family S = 0 is x2 + y2 – 12x – 4y + 38 = 0 (D) the coordinates of point C are (7, 1)

2.20

Let x, y be real variable satisfying the x2 + y2 + 8x – 10y – 40 = 0. Let a = max {(x + 2)2 + (y – 3)2} and b = min {(x + 2)2 + (y – 3)2}, then (A) a + b = 18

2.21

(D) a. b = 73

(B) (1  2 2 , 0)

(C) (4, 1)

(D) (1 , 2 2 )

Point M moved on the circle (x − 4) 2 + (y − 8)2 = 20. Then it broke away from it and moving along a tangent to the circle, cuts the x−axis at the point (− 2, 0). The co− ordinates of a point on the circle at which the moving point broke away is

 3 46  ,   5 5

(A)   2.23

(C) a – b = 4 2

Coordinates of the centre of a circle, whose radius is 2 unit and which touches the line pair x2 – y2 – 2x + 1 = 0, are (A) (4, 0)

2.22

(B) a + b = 4 2

 2 44  ,   5 5

(B)  

(C) (6, 4)

(D) (3, 5)

If the area of the quadrilateral formed by the tangents from the origin to the circle x2 + y2 + 6x –10y + c = 0 and the radii corresponding to the points of contact is 15, then values of c is/ are (A) 9 (B) 4 (C) 5 (D) 25

SECTION - III : ASSERTION AND REASON TYPE 2.24

Statement-1 : Number of common tangents of x2 + y2 – 2x – 4y – 95 = 0 and x2 + y2 – 6x – 8y + 16 = 0 is zero. Statement-2 : If C1C2 < |r1 – r 2|, then there will be no common tangent . (where C1, C2 are the centre and r1 , r2 are radii of circles) (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

2.25

Statement-1 : Let

S1 : x2 + y 2 – 10x – 12y – 39 = 0 S2 : x2 + y2 – 2x – 4y + 1 = 0 and S3 : 2x2 + 2y2 – 20x – 24y + 78 = 0 The radical centre of these circles taken pairwise is (–2, –3) Statement-2 : Point of intersection of three radical axis of three circles taken in pairs is known as radical centre (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 10

 JEE (ADVANCED) - RRB 

2.26

Statement-1 : The equations of the straight lines joining origin to the points of intersection of x2 + y2 – 4x – 2y = 4 and x 2 + y2 – 2x – 4y – 4 = 0 is (y – x)2 = 0 Statement-2 : y + x = 0 is a common chord of x2 + y2 – 4x – 2y = 4 and x2 + y2 – 2x – 4y – 4 = 0 (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

2.27

Statement-1 :Two orthogonal circles intersect to generate a common chord which subtends complimentary angles at their circumferences. Statement-2 : Two orthogonal circles intersect to generate a common chord which subtends supplementary angle at their centres (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

2.28

Statement-1 : For two non-intersecting circles, direct common tangents subtends a right angle at either of point of intersection of circles with line segment joining the centres of circles. Statement-2 : If distance between the centres is more than sum of radii, then circles are non-intersecting (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

SECTION - IV : COMPREHENSION TYPE Read the following comprehensions carefully and answer the questions. Comprehension # 1 Let α-chord of a circle be that chord of the circle which subtends an angle α at the centre. 2.29

If x + y = 1 is α-chord of x2 + y2 = 1, then α is equal to (A)

2.30

2.31

 4

(B)

 2

(C)

 6

(D) x + y = 1 is not a chord

If slope of a

 - chord of x2 + y2 = 4 is 1, then its equation is3

(A) x – y +

6 =0

Distance of

2 - chord of x2 + y2 + 2x + 4y + 1 = 0 from the centre, is 3

(B) x – y = 2 3

(C) x – y =

3

(D) x – y +

3 =0

1

(A) 1

(B) 2

(C)

2

(D)

2

Comprehension # 2 A system of circles is said to be coaxial when every pair of the circles has the same radical axis. It follows from this definition that : 1. The centres of all circles of a coaxial system lie on one straight line, which is perpendicular to the common radical axis. 2. Circles passing through two fixed points form a coaxial system for which the line joining the fixed points is the common radical axis. 3. The equation to a coaxial system, of which two members are S 1 = 0 and S 2 = 0, is S 1 + λS 2 = 0, λ is parameter. If we choose the line of centres as x-axis and the common radical axis as y - axis, then the simplest form of equation of coaxial circles is x2 + y2 + 2gx + c = 0 ...(1) where c is fixed and g is arbitrary. If g = ±

c , then the radius

g2  c vanishes and the circles become point circles. The points

(± c , 0) are called the limiting points of the system of coaxial circles given by (1). 11

 JEE (ADVANCED) - RRB 

2.32

The equation of the circle which belongs to the coaxial system of circles for which the limiting points are (1, – 1), (2, 0) and which passes through the origin is (A) x2 + y 2 – 4x = 0 (B) x2 + y2 + 4x = 0 (C) x2 + y2 – 4y = 0 (D) x2 + y2 + 4y = 0

2.33

If origin be a limiting point of a coaxial system one of whose member is x2 + y2 – 2αx – 2βy + c = 0, then the other limiting point is

 c c , 2 (A)  2 2   2  

   

 c c   , 2 (B)  2 2 2         

  c   , 2 (C)  2 2 2         

 c c   , 2 (D)   2 2 2         

2.34

The equation of the radical axis of the system of coaxial circles x2 + y2 + 2ax + 2by + c + 2λ(ax – by + 1) = 0 is(A) ax – by + 1 = 0 (B) bx + ay – 1 = 0 (C) 2(ax + by) + 1 = 0 (D) 2(bx – ay) + 1 = 0 Comprehension # 3 Two variable chords AB and BC of a circle x2 + y2 = a2 are such that AB = BC = a, and M and N are The mid points of AB and BC respectively such that line joining MN intersect the circle at P and Q where P is closer to AB and O is the centre of the circle

2.35

2.36

2.37

∠OAB is (A) 30°

(B) 60°

Angle between tangents at A and C is (A) 90° (B) 120°

(C) 45°

(D) 15°

(C) 60°

(D) 150°

Locus of point of intersection of tangents at A and C is (A) x2 + y 2 = a2 (B) x2 + y2 = 2a2 (C) x2 + y2 = 4a2

(D) x2 + y2 = 8a2

Comprehension # 4 P is a variable point on the line L = 0. Tangents are drawn to the circle x2 + y2 = 4 from P to touch it at Q and R. The parallelogram PQSR is completed. 2.38

2.39

If L ≡ 2x + y – 6 = 0, then the locus of circumcentre of ΔPQR is (A) 2x – y = 4 (B) 2x + y = 3 (C) x – 2y = 4 If P ≡ (6, 8), then the area of ΔQRS is (A)

2.40

(D) x + 2y = 3

( 6) 3 / 2 sq. units 25

(B)

(24)3 / 2 sq. units 25

(C)

48 6 sq. units 25

(D)

196 6 sq. units 25

If P ≡ (3, 4), then coordinate of S is 63   46 ,  (A)   25   25

 51 68  ,  (B)   25   25

68   46 ,  (C)   25   25

51   68 ,  (D)   25   25

12

 JEE (ADVANCED) - RRB 

SECTION - V : MATRIX - MATCH TYPE 2.41

Column – Ι

Column – ΙΙ

(A)

If ax + by – 5 = 0 is the equation of the chord of the circle (x – 3)2 + (y – 4)2 = 4, which passes through (2, 3) and at the greatest distance from the centre of the circle, then |a + b| is equal to -

(p)

6

(B)

Let O be the origin and P be a variable point on the circle x2 + y2 + 2x + 2y = 0. If the locus of mid-point of OP is x2 + y2 + 2gx + 2fy = 0, then the value of (g + f) is equal to -

(q)

3

(C)

The x-coordinates of the centre of the smallest circle which cuts the circle x2 + y2 – 2x – 4y – 4 = 0 and x2 + y2 – 10x + 12y + 52 = 0 orthogonally, is -

(r)

2

(D)

If θ be the angle between two tangents which are drawn to the

(s)

1

(t)

4 Column – II (p) 1

circle x2 + y2 – 6 3 x – 6y + 27 = 0 from the origin, then

2 3 tan θ equals to 2.42

Column – I (A) The length of the common chord of two circles of radii 3 and

(B)

k , then k equals to 5 The circumference of the circle x2 + y2 + 4x + 12y + p = 0 is bisected by the circle x2 + y2 − 2x + 8y − q = 0, then p + q is equal to -

(q)

24

(C)

Number of distinct chords of the circle 2x ( x  2 ) + y(2y – 1) = 0

(r)

32

(s)

2

(t)

36

4 units which intersect orthogonally is

(D)

1  passing through the point  2 ,  and are bisected 2  by x-axis, is One of the diameters of the circle circumscribing the rectangle ABCD is 4y = x + 7. If A and B are the points (–3, 4) and (5, 4) respectively, then the area of the rectangle is equal to -

SECTION - VI : INTEGER TYPE 2.43

If C1 : x2 + y2 = (3 + 2 2 )2 be a circle and PA and PB are pair of tangents on C1 where P is any point on the director circle of C1, then the radius of smallest circle which touches C1 externally and also the two tangents PA and PB, is -

2.44

A circle touches the hypotenuse of a right angled triangle at its middle point and passes through the middle point of shorter side. If 3 unit and 4 unit be the length of the sides and ‘r’ be the radius of the circle, then find the value of ‘3r’.

2.45

A circle with centre in the first quadrant is tangent to y = x + 10, y = x – 6 and the y-axis. Let (h, k) be the centre of the circle. If the value of (h + k) = a + b a , where (a, b ∈ Q), find the value of (a + b).

2.46

S is a circle having centre at (0, a) and radius b(b < a). A variable circle centred at (α, 0) and touching circle S, meets the X-axis at M and N. A point P ≡  0,   a2  b 2  on the Y-axis, such that ∠MPN is a constant   for any choice of α, then find λ 13

 JEE (ADVANCED) - RRB 

TOPIC

PARABOLA

3

SECTION - I : STRAIGHT OBJECTIVE TYPE 3.1

A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola y2 = 4ax. If PQ is the common chord of the circle and the parabola and L1 L2 is the latus rectum, then the area of the trapezium PL1 L2Q is (A) 3 2 a2

3.2

(B) 2 2 a2

(C) 4 a2

2  2 2  a  2 

(D) 

From the point (15, 12) three normals are drawn to the parabola y2 = 4x, then centroid of triangle formed by three co-normal points is   16 (A)  , 0   3 

(B) (4, 0)

  26 (C)  , 0   3 

(D) (6, 0)

3.3

Through the vertex O of the parabola y2 = 4ax two chords OP & OQ are drawn and the circles on OP & OQ as diameter intersect in R. If θ 1, θ 2 & φ are the angles made with the axis by the tangents at P & Q on the parabola & by OR, then cot θ 1 + cot θ 2 is equal to (A) − 2 tan φ (B) − 2 tan (π − φ) (C) 0 (D) 2 cot φ

3.4

A ray of light travels along a line y = 4 and strikes the surface of a curve y2 = 4(x + y) then equation of the line along reflected ray travels, is (A) x = 0 (B) x = 2 (C) x + y = 4 (D) 2x + y = 4

3.5

If P be a point on the parabola y2 = 3(2x – 3) and M is the foot of perpendicular drawn from P on the directrix of the parabola, then length of each side of an equilateral triangle SMP, where S is focus of the parabola, is (A) 2 (B) 4 (C) 6 (D) 8

3.6

If the locus of middle point of point of contact of tangent drawn to the parabola y2 = 8x and foot of perpendicular drawn from its focus to the tangent is a conic then length of latusrectum of this conic is (A) 9/4 (B) 9 (C) 18 (D) 9/2

3.7

Normals at three points P, Q, R at the parabola y2 = 4ax meet in a point A and S be its focus, if |SP| . |SQ| . |SR| = λ(SA) 2 , then λ is equal to (A) a3 (B) a2 (C) a (D) 1

3.8

If the chord of contact of tangents from a point P to the parabola y2 = 4ax touches the parabola x2 = 4by, the locus of P is (A) circle (B) parabola (C) ellipse (D) hyperbola

3.9

Minimum area of circle which touches the parabola's y = x2 + 1 and y2 = x – 1 is (A)

3.10

3.11

9 sq. unit 16

(B)

9 sq. unit 32

(C)

9 sq. unit 8

(D)

9 sq. unit 4

Let P and Q be points (4, – 4) and (9, 6) of the parabola y2 = 4a(x – b). Let R be a point on the arc of the parabola between P & Q. Then the area of ΔPRQ is largest when (A) ∠PRQ = 90° (B) the point R is (4, 4) 1  (C) the point R is  , 1 (D) None of these 4   If a focal chord of y2 = 4ax makes an angle α, α ∈  0,  with the positive direction of x - axis, then minimum  4 length of this focal chord is (A) 4a (B) 6a (C) 8a (D) None of these 14

 JEE (ADVANCED) - RRB 

2

3.12

Normals AO, AA1, AA2 are drawn to parabola y = 8x from the point A (h, 0) . If triangle OA1A2 (O being the origin) is equilateral, then possible value of ‘h’ is (A) 26 (B) 24 (C) 28 (D) 22

3.13

If the lines (y – b) = m1 (x + a) and y – b = m2 (x + a) are the tangents to y2 = 4ax, then (A) m1 + m2 = 0 (B) m1m2 = 1 (C) m1 + m2 = 1 (D) m1m2 = –1

3.14

The parabola y2 = 4x and circle (x – 6)2 + y2 = r2 will have no common tangent if ‘r’ is (A) r > 20

(B) r <

20

(C) r >

18

(D) r ∈ ( 20 ,

28 )

3.15

Area of the triangle formed by the tangents at the points (4, 6), (10, 8) and (2, 4) on the parabola y2 – 2x = 8y –20, is (in square units) (A) 4 (B) 2 (C) 1 (D) 8

3.16

If P(– 3, 2) is one end of the focal chord PQ of the parabola y2 + 4x + 4y = 0, then the slope of the normal at Q is (A) –

3.17

1 2

S1 : S2 : S3 :

(B) 2

(C)

1 2

From a point (4, 0) three distinct normals can be drawn to the parabola y2 = 8x. Centroid of a triangle formed by joining the foot of three co-normal points on the parabola y2 = 4(x + y) lies on x-axis. The angle between the tangents drawn from the origin to the parabola (x – a)2 = – 4a (y + a), is

 1 tan–1   3 S4 : x + y = 9 is a normal to the parabola y2 = 12x. (A) TFTT (B) TFFT (C) FFTT

3.18

S1 : S2 : S3 :

3.19

S1 : S2 : S3 : S4 :

(D) – 2

(D) FFFT

Vertex of a parabola bisects the subtangent. Subnormal of a parabola is equal to its latusrectum. Circle with focal radius of a point on parabola as diameter touches the tangent drawn at the vertex of the parabola. S4 : Directrix of a parabola is the tangent of a circle drawn its focal chord as diameter. (A) FTTT (B) FFTT (C) TTTT (D* TFTT y = 2x + c is a tangent to the parabola y2 = 4(x + 2) if c = 1/2 Point of contact of tangent y = 2x + c drawn to the parabola y2 = 4(x + 2) is (–7/4, 1) Angle between the tangents drawn from a point (–3, 3) to the parabola y2 = 4(x + 2) is 90º Chord of contact of the parabola y2 = 4(x + 2) drawn from any point on the line x + 3 = 0 passes through the point (– 1, 0) (A) TTTF (B) FTTT (C) TFTF (D) TTFF

SECTION - II : MULTIPLE CORRECT ANSWER TYPE 3.20

Let V be the vertex and L be the latusrectum of the parabola x2 = 2y + 4x – 4. Then the equation of the parabola whose vertex is at V, latusrectum is L/2 and axis is perpendicular to the axis of the given parabola. (A) y2 = x – 2 (B) y2 = x – 4 (C) y2 = 2 – x (D) y2 = 4 – x

3.21

If equation of tangent at P, Q and vertex A of a parabola are 3x + 4y – 7 = 0, 2x + 3y – 10 = 0 and x – y = 0 respectively, then

3.22

(A) focus is (4, 5)

(B) length of latusrectum is 2 2

(C) axis is x + y – 9 = 0

9 9 (D) vertex is  ,  2 2

If A & B are points on the parabola y 2 = 4ax with vertex O such that OA perpendicular to OB & having lengths r1 & r2 respectively, then the value of (A) 16a2

(B) a2

r14 / 3 r24 / 3 is r12 / 3  r22 / 3 (C) 4a

(D) None of these 15

 JEE (ADVANCED) - RRB 

2

3.23

Let P, Q and R are three co-normal points on the parabola y = 4ax. Then the correct statement(s) is/ are (A) algebraic sum of the slopes of the normals at P, Q and R vanishes (B) algebraic sum of the ordinates of the points P, Q and R vanishes (C) centroid of the triangle PQR lies on the axis of the parabola (D) circle circumscribing the triangle PQR passes through the vertex of the parabola

3.24

The locus of the mid point of the focal radii of a variable point moving on the parabola, y2 = 4ax is a parabola whose (A) Latus rectum is half the latus rectum of the original parabola (B) Vertex is (a/2, 0) (C) Directrix is y-axis (D) Focus has the co-ordinates (a,0)

SECTION - III : ASSERTION AND REASON TYPE 3.25

3.26

Statement 1 : If straight line x = 8 meets the parabola y2 = 8x at P & Q then PQ subtends a right angle at the origin. Statement 2 : Double ordinate equal to twice of latus rectum of a parabola subtends a right angle at the vertex. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True Statement 1 : Circumcircle of a triangle formed by the lines x = 0, x + y + 1 = 0 & x – y + 1 = 0 also passes through the point (1, 0) Statement 2 : Circumcircle of a triangle formed by three tangents of a parabola passes through its focus. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

3.27

Statement 1 : Length of focal chord of a parabola y2 = 8x making on angle of 60º with x–axis is 32. Statement 2 : Length of focal chord of a parabola y2 = 4ax making an angle α with x-axis is 4a cosec2α (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

3.28

Statement 1 : Area of triangle formed by pair of tangents drawn from a point (12, 8) to the parabola y2 = 4x and their corresponding chord of contact is 32 sq. units. Statement 2 : If from a point P(x1, y1) tangents are drawn to a parabola y2 = 4ax then area of triangle 3

( y 2  4ax1 ) 2 formed by these tangents and their corresponding chord of contact is 1 sq. 4a units. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

16

 JEE (ADVANCED) - RRB 

3.29

STATEMENT-1 : The perpendicular bisector of the line segment joining the point (–a, 2at) and (a, 0) is tangent to the parabola y2 = 4ax, where t ∈ R STATEMENT-2 : Number of parabolas with a given point as vertex and length of latus rectum equal to 4, is 2. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

3.30

STATEMENT-1 : Normal chord drawn at the point (8, 8) of the parabola y2 = 8x subtends a right angle at the vertex of the parabola. STATEMENT-2 : Every chord of the parabola y2 = 4ax passing through the point (4a, 0) subtends a right angle at the vertex of the parabola. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

SECTION - IV : COMPREHENSION TYPE Read the following comprehensions carefully and answer the questions. Comprehension # 1 y = f(x) is a parabola of the form y = x2 + ax + 1, its tangent at the point of intersection of y-axis and parabola also touches the circle x2 + y2 = r2 . It is known that no point of the parabola is below x-axis. 3.31

The radius of circle, when 'a' attains its maximum value, is 1

(A)

3.32

3.33

10

1

(B)

5

(C) 1

(D)

The slope of the tangent, when radius of the circle is maximum, is (A) 0 (B) 1 (C) – 1

5

(D) not defined

The minimum area bounded by the tangent and the coordinate axes (A)

1 4

(B)

1 3

(C)

1 2

(D) 1

Comprehension # 2 If the locus of the circumcentre of a variable triangle having sides y-axis, y = 2 and x + my = 1, where (, m) lies on the parabola y2 = 4ax is a curve C, then 3.34

Coordinates of the vertex of this curve C is 3  (A)  2a,  2 

3.35

3  (B)   2a,   2 

3  (C)   2a,  2 

3  (D)   2a,   2 

The length of smallest focal chord of this curve C is : (A)

1 12a

(B)

1 4a

(C)

1 16a

(D)

1 8a 17

 JEE (ADVANCED) - RRB 

3.36

The curve C is symmetric about the line : (A) y = –

3 2

(B) y =

3 2

(C) x = –

3 2

(D) x =

3 2

Comprehension # 3 In general, three normals can be drawn from a point to a parabola and the point where they meet the parabola are called co-normal points. The equation of any normal to y2 = 4ax is y = mx – 2am – am3. If it passes through (h, k), then k = mh – 2am – am3 or am 3 + m(2a – h) + k = 0 This is cubic in m, it has three roots m1, m2, m3 ∴ m1 + m2 + m3 = 0, m 1m2 m3 = 3.37

Minimum distance between the curves y2 = x – 1 and x2 = y – 1 is equal to (A)

3.38

3 2 4

(B)

5 2 4

(C)

7 2 4

(D)

2 4

If the normals from any point to the parabola x2 = 4y cuts the line y = 2 in points whose abscissaes are in A.P., then the slopes of the tangents at the three co-normal points are in (A) A.P.

3.39

k 2a  h , m 1m2 + m2m 3 + m3m 1 = a a

(B) G.P.

(C) H.P.

(D) None of these

If the normals at three points. P, Q, R of the parabola y2 = 4ax meet in a point O and S be its focus, then |SP|.|SQ|.|SR| is equal to (A) a2 (B) a(SO)3 (C) a(SO)2 (D) None of these

SECTION - V : MATRIX - MATCH TYPE 3.40

Column – Ι

Column – ΙΙ

(A)

Area of a triangle formed by the tangents drawn from a point (–2, 2) to the parabola y2 = 4(x + y) and their corresponding chord of contact is

(p)

8

(B)

Length of the latusrectum of the conic 25{(x – 2)2 + (y – 3)2} = (3x + 4y – 6) 2 is

(q)

4 3

(C)

If focal distance of a point on the parabola y = x2 – 4 is

(r)

4

(s)

12 5

(t)

24 5

25/4 and points are of the form (±

a , b) then value of

a + b is

(D)

Length of side of an equilateral triangle inscribed in a parabola y2 – 2x – 2y – 3 = 0 whose one angular point is vertex of the parabola, is

18

 JEE (ADVANCED) - RRB 

3.41

Column – Ι

Column – ΙΙ

(A)

Parabola y2 = 4x and the circle having its centre at (6, 5) intersects at right angle, at the point (a, a) then one value of a is equal to

(B)

The angle between the tangents drawn to (y – 2)2 = 4(x + 3) (q) at the points where it is intersected by the line 3x – y + 8 = 0 is

(C)

(p)

13

8

4 , then p has the value equal to p

If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the value of k is

(r)

10 5

Length of the normal chord of the parabola y2 = 8x at the point (s) 4 where abscissa & ordinate are equal is (t) 12 Column – Ι Column – ΙΙ (D)

3.42

(A)

Radius of the largest circle which passes through the focus of the parabola y2 = 4x and contained in it , is

(p)

16

(B)

Two perpendicular tangents PA & PB are drawn to the parabola y 2 = 16x then minimum value of AB is

(q)

5

(C)

The shortest distance between parabolas y2 = 4x and y2 = 2x – 6 is d then d2 =

(r)

8

(D)

The harmonic mean of the segments of a focal chord of the parabola y2 = 8x

(s)

4

(t)

12

SECTION - VI : INTEGER ANSWER TYPE 3.43

The chord of the parabola y2 = 4 a x, whose equation is y − x 2 + 4a

2 = 0, is a normal to the curve,

and its length is λ 3 a , then find λ.

3.44

The two parabolas y2 = 4ax and y2 = 4(a – 1) (x – b) can not have common normal other than axis unless b > λ, then find λ.

3.45

Tangents and normals at points P and Q to the parabola y2 = 4x intersect at point T and point R (9, 6) respectively. Then find the length of tangent drawn from (–1, –1) to the circle circumscribing the quadrilateral PTQR .

3.46

From a point A common tangents are drawn to the circle x2 + y2 = a2 /2 and the parabola y 2 = 4a x. Find the area of the quadrilateral formed by the common tangents, the chords of contact of the point A, w.r.t. the circle and the parabola is

a2 , then find λ 4

19

 JEE (ADVANCED) - RRB 

TOPIC

ELLIPSE

4

SECTION - I : STRAIGHT OBJECTIVE TYPE x2 y2 + = 1 at any point P meet the line x = 0 at a point Q. Let R be the image of 25 16 Q in the line y = x, then circle whose extrimities of a diameter are Q and R passes through a fixed point. The fixed point is (A) ( 3, 0) (B) (5, 0) (C) (0, 0) (D) (4, 0)

4.1

A tangent to the ellipse

4.2

Find the locus of point of intersection of pair of tangents to the ellipse if the sum of the ordinates of the point of contact is b.

4.3

 x2 y2  b   (A)  2  2  4 y = 1 a b  

 x2 y2  b   (B)  2  2  2y = 1 a b  

 x 2 y 2  2b   (C)  2  2  y = 1 b  a

 x2 y2  b   (D)  2  2  2y  4 b  a

An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is : (A) 2

(B) 3

(C)

3

(D) 4

4.4

A ray emanating from the point (0, 6) is incident on the ellipse 25x2 + 16y2 = 1600 at the point P with ordinate 5. After reflection, ray cuts the y-axis at B. Find the length of PB. (A) 7 (B) 13 (C) 5 (D) none of these

4.5

Any ordinate MP of an ellipse

4.6

P & Q are two points on the ellipse, 9 x2 + 25 y2 = 225 such that sum of their ordinates is 3. Find the locus of the point intersection of tangents at P and Q. (A) 9 x2 – 25 y2 − 150 y = 0 (B) 9 x2 + 25 y2 + 150 y = 0 (C) 9 x2 + 25 y2 − 150 y = 2 (D) 9 x2 + 25 y2 − 150 y = 0

4.7

If PQ is focal chord of ellipse

x2 y2   1 meets the auxiliary circle in Q, then locus of point of 25 9 intesection of normals at P and Q to the respective curves, is (B) x2 + y2 = 34 (C) x2 + y2 = 64 (D) x2 + y2 = 15 (A) x2 + y2 = 8

PQ is (A) 8 4.8

x2 y2 + = 1 which passes through S ≡ (3, 0) and PS = 2 then length of chord 25 16

(B) 6

(C) 10

(D) 4

If P is a moving point in the xy–plane in such a way that perimeter of triangle PQR is 16 {where Q ≡ (3, 5 ), R ≡ (7, 3 5 )} then maximum area of triangle PQR is (A) 6 sq. unit (B) 12 sq. unit (C) 18 sq. unit (D) 9 sq. unit

20

 JEE (ADVANCED) - RRB 

4.9

If f(x) is a decreasing function then the set of values of ‘k’, for which the major axis of the ellipse

x2 f (k  2k  5) 2

+

y2 = 1 is the x-axis, is: f (k  11)

(A) k ∈ (–2, 3) (C) k ∈ (–∞, –3) U (2, ∞)

4.10

(B) k ∈ (–3, 2) (D) k ∈ (–∞, –2) U (3, ∞) x2 y2 + =1 16 9

The equation to the locus of the middle point of the portion of the tangent to the ellipse

included between the co-ordinate axes is the curve : (A) 9x2 + 16y2 = 4 x2y2 (B) 16x2 + 9y2 = 4 x2y2 (C) 3x2 + 4y2 = 4 x2y2 (D) 9x2 + 16y2 = x2 y2

4.11

The line, x + my + n = 0 will cut the ellipse if: (A) a22 + b2n2 = 2 m 2

4.12

x2 y 2  2 + 2 = 1 in points whose eccentric angles differ by 2 a b

(B) a2m 2 + b22 = 2 n2

(C) a22 + b2m 2 = 2 n2

(D) a2n2 + b2m 2 = 2 2

x2 y2 Q is a point on the auxiliary circle corresponding to the point P of the ellipse 2  2 = 1. If T is the a b foot of the perpendicular dropped from the focus S onto the tangent to the auxiliary circle at Q then the ΔSPT is: (A) isosceles (B) equilateral (C) right angled (D) right isosceles

4.13

The angle between the pair of tangents drawn to the ellipse, 3x2 + 2y2 = 5 from the point (1, 2) is:

 6    5

 

(B) arc tan 

(A) arc tan 6 5

4.14

 12    5



(C) arc tan 

(D) arc tan 12 5

The equation to the locus of the middle point of the portion of the tangent to the ellipse included between the co-ordinate axes is the curve: (A) 9x2 + 16y2 = 4 x2y2 (B) 16x2 + 9y2 = 4 x2y2 (C) 3x2 + 4y2 = 4 x2y2 5

4.15

A circle of radius r =

2

is concentric with the ellipse

by the common tangent with the line (A)

4.16

 3

(B)

 x2 y2 + =1 16 9

(D) 9x2 + 16y 2 = x2y2

x2 y2 + = 1. Then find the acute angle made 16 9

3 x  y  6 = 0 is -

 6

(C)

 4

(D)

 12

x2 y2 The tangent at the point 'α' on the ellipse 2 + 2 = 1 meets the auxiliary circle in two points which a b subtends a right angle at the centre, then the eccentricity 'e' of the ellipse is given by the equation: (A) e2 (1 + cos2 α) = 1 (B) e2. (cosec2 α − 1) = 1(C) e 2 (1 + sin2 α) = 1 (D) e2 (1 + tan2 α) = 1

4.17

If circumcentre of an equilateral triangle inscribed in

x2

y2

= 1, with vertices having eccentric a2 b2 angles α, β, γ respectively is (x1, y1), then Σ cosα cosβ + Σ sin α sin β = 2

(A)

9 x1 a

2

2

+

9 y1 b

2

+

3 (B) 9x12 – 9y1 2 + a2 b2 2

2

(C)

+

2

2

9 x1 9 y1 + a b

+3

(D)

9 x1 2a

2

2

+

9 y1 2b

2

–

3 2 21

 JEE (ADVANCED) - RRB 

4.18

A series of concentric ellipses E1, E2, ......, En are drawn such that En touches the extremities of the major axis of En –1 and the foci of En coincide with the extremities of minor axis of En – 1. If the eccentricity of the ellipses is independent of n, then the value of the eccentricity, is (A)

4.19

5 3

5 1 2

(B)

(C)

5 1 2

1

(D)

5

S1 : Length of the latus rectum of the ellipse x2 + 4y2 – 2x – 16y + 13 = 0 is 1. S2 : Distance between foci of the ellipse x2 + 4y2 – 2x – 16y + 13 = 0 is 4 3 . S3 : Sum of the focal distances of a point P(x, y) on the ellipse x2 + 4y2 – 2x – 16y + 13 = 0 is 4. S4 : y = 3 meets the tangents drawn at the vertices of the ellipse x2 + 4y2 – 2x – 16y + 13 = 0 at points P & Q then PQ subtands a right angle at any of its foci. (A) TFTT (B) TTTT (C) TFFT (D) TFTF

4.20

S1 : A ray of light emanating from a point (3, 0) and strikes the positive end of minor axes of the ellipse 16x2 + 25y2 = 400, then after reflection from minor axis, it travels along the line whose slope is 3/4. S2 : Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centre is π/4. S3 : Eccentricity of the ellipse 9x2 + 5y2 – 30y = 0 is 2/3 S4 : Centroid of the triangle of greatest area inscribed in the ellipse is either (0, 1) or (0, –1). (A) FTTF (B) FTTT

4.21

x2 y2 + = 1 taking major axis as base 36 9

(C) TTTT

(D) FFTT

S1 : If from a point P(0, α) two normals other than axes are drawn to ellipse

x2 y2  = 1, 25 16

9 4 S2 : The minimum and maximum distances of a point (1, 2) from the ellipse 4x2 + 9y2 + 8x – 36y + 4 = 0 are L and G, then G – L is equal to 4

where |α| ≤ k, then least value of k is

S3 : If the length of latus rectum of an ellipse is one-third of its major axis. Its eccentricity is equal to

2 3

2

 5 x  12 y  1   represents an ellipse S4 : The set of all positive values of a for which (13x – 1)2 + (13y – 2)2 =  a   is (1, 2) (A) TTFF (B) TTFT (C) TTTF (D) TFTF

SECTION - II : MULTIPLE CORRECT ANSWER TYPE 4.22

4.23

If P is a point of the ellipse

x2 a2

+

y2 b2

then (A)

PS + PS′ = 2a, if a > b

(C)

tan

1 e   tan = 1 e 2 2

(D)

tan

  a2  b2 tan = [a – 2 2 b2

= 1, whose focii are S and S′. Let ∠PSS′ = α and ∠PS′S = β, (B)

a 2  b2 ]

PS + PS′ = 2b, if a < b

when a > b

The parametric angle θ, where –π – θ ≤ π, of the point on the ellipse

x2



y2

a2 b 2 drawn cuts the intercept of minimum length on the coordinate axes, is/are

(A) tan–1

b a

(B) –tan–1

b a

(C) π – tan–1

b a

 1 at which the tangent

(D) π + tan–1

b a 22

 JEE (ADVANCED) - RRB 

4.24

The equation, 3x + 4y − 18x + 16y + 43 = c. (A) cannot represent a real pair of straight lines for any value of c (B) represents an ellipse, if c > 0 (C) represent empty set, if c < 0 (D) a point, if c = 0

4.25

Let F1, F2 be two focii of the ellipse and PT and PN be the tangent and the normal respectively to the ellipse at point P then (A) PN bisects ∠ F1 PF 2 (B) PT bisects ∠F 1PF2 (C) PT bisects angle (180° – ∠ F1PF2) (D) None of these

4.26

Let A(α) and B(β) be the extremeties of a chord of an ellipse . If the slope of AB is equal to the slope of the tangent at a point C(θ) on the ellipse, then the value of θ, is

2

(A)

  2

2

(B)

  2

(C)

 +π 2

(D)

  –π 2

SECTION - III : ASSERTION AND REASON TYPE 4.27

4.28

Statement-1 : Locus of centre of a variable circle touching two circles (x – 1)2 + (y – 2)2 = 25 and (x – 2)2 + (y – 1)2 = 16 is an ellipse. Statement-2 : If a circle S2 = 0 lies completely inside the circle S1 = 0 then locus of centre of a variable circle S = 0 which touches both the circles is an ellipse. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

3 3    Statement-1: If P  2 ,1 is a point on the ellipse 4x2 + 9y2 = 36. Circle drawn taking AP as diameter   touches another circle x2 + y2 = 9, where A ≡(– 5 , 0). Statement-2 : Circle drawn with focal radius as diameter touches the auxiliary circle. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

4.29

Statement-1 : Feet of perpendiculars drawn from foci of an ellipse 4x2 + y2 = 16 on the line 2 3 x + y = 8 lie on the circle x2 + y2 = 16. Statement-2 : If perpendicular are drawn from foci of an ellipse to its any tangent then feet of these perpendiculars lie on director circle of the ellipse. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

4.30

Statement-1 : In a triangle ABC, if base BC is fixed and perimeter of the triangle is constant, then vertex A moves on an ellipse. Statement-2 : If sum of distances of a point ‘P’ from two fixed points is constant then locus of ‘P’ is a real ellipse. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

23

 JEE (ADVANCED) - RRB 

4.31.

STATEMENT-1 : Let tangent at a point P on the ellipse, which is not an extremity of major axis, meets a directrix at T. If circle drawn on PT as diameter cuts the directrix at Q, then PQ = ePS, where S is the focus corresponding to the directrix. STATEMENT-2 : Let tangent at a point P on an ellipse, which not an extremity of major axis, meets the directrices at T′ and T. Then PT substends a right angle at the focus corresponding the directrix at which T lies. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

4.32

STATEMENT-1 : A triangle ABC right angled at A moves so that its perpendicular sides touch the curve

x2 y 2 + = 1 all the time. Then loci of the points A, B and C are circle. a 2 b2 STATEMENT-2 : Locus of point of intersection of two perpendicular tangents to the conic is director circle (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

SECTION - IV : COMPREHENSION TYPE Read the following comprehensions carefully and answer the questions. Comprehension # 1 An ellipse E has its centre C (1, 3) focus at S(6, 3) and passing through the point P(4, 7), then 4.33

The product of the lengths of the perpendicular segments from the focii on tangent at point P is (A) 20 (B) 45 (C) 40 (D) can not be determined

4.34

The point of intersection of the lines joining each focus to the foot of the perpendicular from the other focus upon the tangent at point P, is 5  (A)  , 5  3 

4.35

4  (B)  , 3  3 

8  (C)  , 3  3 

 10  (D)  , 5   3 

If the normal at a variable point on the ellipse (E) meets its axes in Q and R then the locus of the mid point of QR is a conic with an eccentricity (e′), then 3

(A) e′ =

10

(B) e′ =

5 3

3

(C) e′ =

5

(D) e′ =

10 3

Comprehension # 2 Consider an ellipse (E)

x2



y2

= 1, centered at point ‘O’ and having AB and CD as its major and a2 b2 minor axes respectively if S1 be one of the focii of the ellipse, radius of incircle of triangle OCS1, be 1 unit and OS1 = 6 units, then

4.36

The area of ellipse (E) is (A)

4.37

65  4

(B)

64  5

Perimeter of ΔOCS, is (A) 20 units (B) 10 units

(C) 64 π

(D) 65 π

(C) 15 units

(D) 25 units 24

 JEE (ADVANCED) - RRB 

4.38

If S be the director circle of ellipse (E) their the equation of director circle of S is (A) x2 + y2 = (48.5)

(B) x2 + y 2 =

97

(C) x2 + y2 = 97 (D) x2 + y 2 =

48.5

Comprehension – 3 Second degree equation ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0 represents an ellipse if

a h g h b f  0 & h 2 < ab. Intersection of major axis and minor axis gives centre of ellipse g f

c

4.39

There are exactly ‘n’ integral values of λ for which equation x2 +λ xy + y2 = 1 represents an ellipse then ‘n’ must be __ (A) 0 (B) 1 (C) 2 (D) 3

4.40

Length of the longest chord of the ellipse x2 + y2 + xy = 1 is __ 1

(A) 4.41

(B)

2

2

(C) 2 2

(D) 1

Length of the chord perpendiuclar to longest chord as in above question and pasing through centre of ellipse is __ 1

(A)

(B)

2

3 2

(C) 2

1

2 3

(D)

3

Comprehension – 4 A bird flies on ellipse ax2 + by2 = 1 & z = 5 3 (b > a > 0) whose eccentricity is

1

. An observer stands at 2 a point P(α, β, 0) where maximum and minimum angle of elevation of the bird are 60º and 30º when bird is at Q and R respectively on its path and Q′ and R′ are projection of Q and R on x- y plane, P, Q′ R′ are collinear & the distance between Q′ and R′ is maximum Let θ be the angle elevation of the bird when it is at a point on the arc of the ellipse exactly mid-way between Q & R. It is given that aα2 + bβ2 – 1 > 0 4.42

If α > 0, then equation of the line along which minimum angle of elevation is observed, is (A)

(C)

4.43

x 3 3 x  10 3

z 3 1

(B)

=

z 3 ,y=0 1

(D)

x  13 3 x  10 3

=

z 3 , y=0 1

y z 3 = 0 = 1

Equation of plane which touches the ellipse at Q and passes through P (α > 0) is (A) – (C)

4.44

=

3 x + y + z – 10 3 = 0 3 x + z – 10 3 = 0

(B)

3 x + y + z – 10 3 = 0

(D)

3 x + y – 10 3 = 0

Value of tan θ, is (A)

2 3

(B)

3 2

(C)

2 3

(D)

6 5 25

 JEE (ADVANCED) - RRB 

SECTION - V : MATRIX - MATCH TYPE 4.45

Column – Ι (A)

Column – ΙΙ

A tangent to the ellipse

slope –

y2 x2 + = 1 having 48 27

(p)

36

(q)

72

(r)

10

(s)

16

(t)

10 2

4 cuts the x and y-axis at the points 3

A and B respectively. If O is the origin then area of triangle OAB is equal to (B)

Product of the perpendiculars drawn from the points (± 3, 0) to the line y = mx –

(C)

25m 2  16 is

An ellipse passing through the origin has its foci (3, 4) and (6, 8), then length of its minor axis is

(D)

If PQ is focal chord of ellipse

x2 y2 + = 1 which 25 16

passes through S ≡ (3, 0) and PS = 2 then length of chord PQ is

4.46

Column – Ι

Column – ΙΙ

(A)

(p)

A stick of length 10 meter slides on co-ordinate

6

axes, then locus of a point dividing this stick reckoning from x-axis in the ratio 6 : 4 is a curve whose eccentrictly is e, then 9e is equal to (B)

AA′ is major axis of an ellipse 3x2 + 2y2 + 6x – 4y – 1 = 0

(q) 2 7

& P is a variable point on it then greatest area of triangle APA′ is (C)

Distance between foci of the curve represented by

(r)

128 3

the equation x = 1 + 4 cosθ, y = 2 + 3 sinθ is (D)

Tangents are drawn to the ellipse

x2 y2 + = 1 at 16 7

(s) 3 5

end points of latusrectum. The area of equadrilateral so formed is (t)

5 3

26

 JEE (ADVANCED) - RRB 

4.47

Match the column Column – Ι

Column – ΙΙ

(A)

(p)

4

(q)

2

If the angle between the straight angle lines joining foci and one of the ends of the minor axis of the ellipse

x2 a2



y2 b2

=1

is 90º. Find its eccentricity.

(B)

For an ellipse

x2 y 2   1 with vertices A and A′, tangent 9 4

drawn at the point P in the first quadrant meets the y-axis in Q and the chord A′P meets the y-axis in M. If ‘O’ is the origin, then OQ2 – MQ2 equals to 1

(C)

The x-coordinate of points on the axis of the parabola

(r)

2

4y2 – 32x + 4y + 65 = 0 from which all the three normals to the parabola are real is (D)

The area of the parallelogram inscribed in the ellipse

x2 22



y2 (1/ 2)2

(s)

7

(t)

8

= 1 whose diagonals are the conjugate

diameters of the ellipse is given by

SECTION - VI : INTEGER ANSWER TYPE x2 y2  = 1 from the point P (0, 6) is 169 25

4.48

Number of distinct normal lines that can be drawn to ellipse

4.49

Origin O is the centre of two concentric circles whose radii are a & b respectively, a < b. A line OPQ is drawn to cut the inner circle in P & the outer circle in Q. PR is drawn parallel to the y− axis & QR is drawn parallel to the x− axis. The locus of R is an ellipse touching the two circles. If the focii of this ellipse lie on the inner circle, if eccentricity is

4.50

Number of points on the ellipse

ellipse

2  , then find λ

x2 y2 + = 1 from which pair of perpendicular tangents may be drawn to the 50 20

x2 y2  = 1 is 16 9

27

 JEE (ADVANCED) - RRB 

TOPIC

HYPERBOLA

5

SECTION - I : STRAIGHT OBJECTIVE TYPE 5.1

From a point P(1, 2) pair of tangent’s are drawn to a hyperbola ‘H’ where the two tangents touch different arms of hyperbola. Equation of asymptotes of hyperbola H are eccentricity of ‘H’ is (A) 2

(B)

2

(C)

3

3x–y+5=0&

2

(D)

3 x + y – 1 = 0 then

3

5.2

The asymptotes of a hyperbola are parallel to 2 x + 3 y = 0 & 3 x + 2 y = 0. Its centre is (1, 2) & it passes through (5, 3).Find the equation of the hyperbola. (A) (2x + 3y – 8) (3x + 2y – 7) + 154 = 0 (B) (2x + 3y + 8) (3x + 2y – 7) – 154 = 0 (C) (2x + 3y – 8) (3x + 2y – 7) – 154 = 0 (D) (2x + 3y – 8) (3x + 2y + 7) – 154 = 0

5.3

If angle between asymptote’s of hyperbola

x2 a2

–

y2 b2

= 1 is 120º and product of perpendiculars drawn

from foci upon its any tangent is 9, then locus of point of intersection of perpendicular tangents of the hyperbola can be – (B) x2 + y2 = 9 (C) x2 + y2 = 3 (D) x2 + y2 = 18 (A) x2 + y2 = 6 5.4

‘C’ be a curve which is locus of point of intersection of lines x = 2 + m and my = 4 – m. A circle S ≡ (x – 2)2 + (y + 1)2 = 25 intesects the curve C at four points P, Q, R and S. If O is centre of the curve ‘C’, then OP2 + OQ2 + OR2 + OS2 is (A) 50 (B) 100 (C) 25 (D) 25/2

5.5

The combined equation of the asymptotes of the hyperbola 2x2 + 5xy + 2y2 + 4x + 5y = 0 is (A) 2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0 (B) 2x2 + 5xy + 2y2 + 4x + 5y – 2 = 0 2 2 (C) 2x + 5xy + 2y = 0 (D) none of these

5.6

If α + β = 3π then the chord joining the points α and β for the hyperbola

x2 a2

–

y2 b2

= 1 passes through

(A) focus (B) centre (C) one of the end points of the transverse axis (D) one of the end points of the conjugates axis

5.7

For a given non-zero value of m each of the lines x2 a2

(A)

–

y2 b2

x y x y – = m and + = m meets the hyperbola a b a b

= 1 at a point. Sum of the ordinates of these points, is

a(1  m 2 ) m

(B)

b (1  m2 ) m

(C) 0

(D)

ab 2m

28

 JEE (ADVANCED) - RRB 

2

5.8

The equation of the transverse axis of the hyperbola (x – 3) + (y + 1) = (4x + 3y)2 is (A) x + 3y = 0 (B) 4x + 3y = 9 (C) 3x – 4y = 13 (D) 4x + 3y = 0

5.9

For which of the hyperbola, we can have more than one pair of perpendicular tangents? (A)

5.10

x2 y2  =1 4 9

5.12

x2 y2  =–1 4 9

(B) Ι & ΙV quadrants

(C) Ι & ΙΙΙ quadrants

(D) ΙΙΙ & ΙV quadrants

y x + =1 x1  x 2 y1  y 2

(B)

y x + =1 x1  x 2 y1  y 2

(C)

y x + =1 y1  y 2 x1  x 2

(D)

y x + =1 y1  y 2 x1  x 2

The locus of the foot of the perpendicular from the centre of the hyperbola xy = c2 on a variable tangent is: (B) (x2 + y2)2 = 2c2 xy (D) (x2 + y2)2 = 4c2 xy

Find the range of parameter a for which a unique circle will pass through the points of intersection of the rectangular hyperbola x2 – y2 = a2 and the parabola y = x2.

If the curves

 1 1 (C)   ,   2 2

(B) (0, 1)

 1 1 (D)   ,   2 4

x2

y2 + = 1, (a > b) and x2 – y2 = c2 cut at right angles then a 2 b2

(A) a2 + b2 = 2c2

5.15

x2 y2  = 1 then point of contacts lie in 16 9

(A)

(A) a ∈ (–1, 1)

5.14

(D) xy = 4

The equation to the chord joining two points (x1, y1 ) and (x2, y2) on the rectangular hyperbola xy = c2 is:

(A) (x2 − y 2)2 = 4c2 xy (C) (x2 + y2) = 4c2 xy

5.13

(C) x2 – y2 = 4

From point (2, 2) tangents are drawn to the hyperbola (A) Ι & ΙΙ quadrants

5.11

(B)

2

(B) b2 – a2 = 2c2

If radii of director circles of

x2 a2

+

y2 b2

= 1 and

(C) a2 – b2 = 2c2

x2 a2

–

y2 (b ) 2

(D) a2 b2 = 2c2

= 1 are 2r and r respectively and ee and e h

be the eccentricities of the ellipse and the hyperbola respectively then (A) 2eh2 – e e2 = 6 (B) ee2 – 4eh2 = 6 (C) 4eh2 – e e2 = 6 (D) none of these

5.16

If the foci of the ellipse

1 x2 y2 x2 y2  2 = 1 & the hyperbola  = coincide then the value of b2 is : 25 b 144 81 25

(A) 4

(B) 9

(C) 16

(D) none 29

 JEE (ADVANCED) - RRB 

5.17

2

The tangent at any point P(x1, y1) on the hyperbola xy = c meets the co-ordinate axes at points Q & R. The circumcentre of ΔOQR has co-ordinates. (B) (x1, y1)

(A) (0, 0)

5.18

(C)

 x1 y1   ,   2 2

(D)

 2x1 2y1  ,    3 3 

The locus of the mid points of the chords passing through a fixed point (α, β) of the hyperbola

x2 y 2 − = 1 is : a 2 b2    ,   2 2

   ,   2 2

(A) a circle with centre 

(B) an ellipse with centre 

   ,   2 2

(C) a hyperbola with centre 

   ,   2 2

(D) straight line passing through 

5.19

If two conics a1x2 + 2 h1xy + b1y2 = c1 and a2x2 + 2 h2xy + b2y2 = c2 intersect in four concyclic points, then (A) (a1 − b1) h2 = (a2 − b2) h1 (B) (a1 − b1) h1 = (a2 − b2) h2 (C) (a1 + b1) h2 = (a2 + b2) h1 (D) (a1 + b1) h1 = (a2 + b2) h2

5.20

The transverse axis of a hyperbola is of length 2a and lies along x axis, a vertex divides the segment of the axis between the centre and the corresponding focus in the ratio 2 : 1, the equation of the hyperbola is : (A) 4x2 – 5y2 = 4a2 (B) 4x2 – 5y2 = 5a2 (C) 5x2 – 4y2 = 4a2 (D) 5x2 – 4y2 = 5a2

5.21

S1 :

Number of points from where perpendicular tangents can be drawn to the hyperbola 16x2 – 9y2 = 144 is infinite.

S2 :

If distance between two parallel tangents drawn to the hyperbola

is equal to ±

S3 :

x2 y2 – = 1 is 2 then their slope 9 49

5 . 2

If through the point (5, 0) chords are drawn to the hyperbola

x2 y2 – = 1. Then locus of their 25 9

middle points is also a hyperbola whose length of latus rectum is same as given hyperbola 9x2 – 25y2 = 225. S4 :

If the line y = mx +

a 2m 2  b 2 touches the hyperbola

x2 a2

–

y2 b2

= 1 at the point

 b  . (a sec θ, b tan θ) then θ = sin–1   am 

(A) TTFT

(B) FTTT

(C) TFFT

(D) FTFT

30

 JEE (ADVANCED) - RRB 

5.22

S1 : S2 : S3 : S4 :

If x = 3 & y = 2 are the equations of asymptotes of a hyperbola and hyperbola passes through the point (4, 6) then length of its latus rectum is 4 2 . Two concentric rectangular hyperbolas whose axes meet at an angle π/4, cut each other at an angle π/2. Distance between directrices of hyperbola xy = 16 is 4 If line joining the points A(x1 0) & B(0, y1) is tangent to the hyperbola xy = c2 then point of contact  x1 y1  is  ,  .  2 2

(A) TTFT 5.23

(B) TFTT

(C) FFTT

(D) FFTF

S1 : S2 :

Centre of the hyperbola x2 – 4y2 – 4x + 8y + 4 = 0 is (2, 1) Product of the length of perpendiculars drawn from any foci of the hyperbola x2 – 4y2 – 4x + 8y + 4 = 0 to its asymptotes is 4.

S3 :

If eccentricity of hyperbola x(y – 1) = 2 is

S4 :

Point (2, 2) lies outside the hyperbola (A) TFFT

2 then eccentricity of its conjugate hyperbola is 2.

x2 y2 – = 1. 16 9

(B) TTFT

(C) TTFF

(D) TFTF

SECTION - II : MULTIPLE CORRECT ANSWER TYPE

5.24

If foci of

x2 a2

–

y2 b2

= 1 concide with the focii of

then (A) a2 + b2 = 16 (C) centre of the director circle is (0, 0)

5.25

The lines y = mx ±

x2 y2 + = 1 and eccentricity of the hyperbola is 2, 25 9

(B) there is no director circle to the hyperbola (D) length of latus ractum of the hyperbola = 12

a 2m 2  b 2 , m > 0 touches the hyperbola

x2 a2

–

y2

= 1 at the point’s whose

b2

eccentric angle is  b   (A) sin–1   ma 

5.26

 b   (B) π + sin–1   ma 

 b   (D) – sin–1   ma 

For the hyperbola 9x2 – 16y 2 – 18x + 32y – 151 = 0 (A) one of the directrix is x =

21 5

(C) Focii are (6, 1) and (–4, 1)

5.27

 b   (C) 2π + sin–1   ma 

(B) Length of latus rectum =

(D) eccentricity is

9 2

5 4

If (a sec θ, b tan θ) and (a secφ, b tan φ) are the ends of a focal chord of

tan

  tan equals to 2 2

(A)

e 1 e 1

(B)

1 e 1 e

(C)

1 e 1 e

(D)

x2 a2

–

y2 b2

= 1, then

e 1 e 1 31

 JEE (ADVANCED) - RRB 

SECTION - III : ASSERTION AND REASON TYPE

5.28

x2 y2 + = 1 and 12x2 – 4y2 = 27 intersect each other at right angle. 25 16

Statement -1 : Ellipse

Statement -2 : Whenever con focal conics intersect, they intersect each other orthogonally. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 5.29

Statement -1 : A bullet is fired and hit a target. An observer in the same plane heard two sounds the crack of the rifle and the thud of the bullet striking the target at the same instant, then locus of the observer is hyperbola where velocity of sound is smallar than velocity of bullet. Statement -2 : If difference of distances of a point ‘P’ from the two fixed points is constant and less than the distance between the fixed points then locus of ‘P’ is a hyperbola. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

5.30

Statement -1 : With respect to a hyperbola

x2 y2 – = 1 pependicular are drawn from a point (5, 0) on the 9 16

lines 3y ± 4x = 0, then their feet lie on circle x2 + y2 = 16. Statement -2 : If from any foci of a hyperbola perpendicular are drawn on the asymptotes of the hyperbola then their feet lie on auxiliary circle. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 2

5.31

Statement -1 : If eccentricity of a hyperbola is 2 then eccentricity of its conjugate hyperbola is

3

.

Statement -2 : If e and e′ are the eccentricities of a hyperbola and its conjugate hyperbola then

1 e

2

+

1 e2

= 1.

(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 5.32

Statement -1 : If a circle S = 0 intersects a hyperbola xy = 4 at four points. Three of them are (2, 2) (4, 1) and (6, 2/3) then co-ordinates of the fourth point are (1/4 , 16). Statement -2 : If a circle S = 0 intersects a hyperbola xy = c2 at t1, t2, t3, t4 , then t1.t2.t3.t4 = 1. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

32

 JEE (ADVANCED) - RRB 

5.33

2

2

Statement -1 : If a tangent is drawn to a hyperbola 16x – 9y = 144 at a point (15/4, 3) then another tangent at the point (–15/4, –3) will be parallel to the previous tangent. Statement -2 : Two parallel tangents to a hyperbola touches the hyperbola at the extremities of a diameter and converse is also true. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

SECTION - IV : COMPREHENSION TYPE Read the following comprehensions carefully and answer the questions. Comprehension # 1 If P is a variable point and F1 and F2 are two fixed points such that |PF1 – PF2| = 2a. Then the locus of the point P is a hyperbola, with points F1 and F2 as the two focii (F 1F2 > 2a). If

hyperbola, then its conjugate hyperbola is

( x  1)2  ( y  2)2  ( x  5)2  ( y  5)2

5.34

a2

–

y2 b2

a2

–

y2 b2

= 1 is a

= – 1. Let P(x, y) is a variable point such that

= 3.

If the locus of the point P represents a hyperbola of eccentricity e, then the eccentricity e′ of the corresponding conjugate hyperbola is : (A)

5.35

x2

x2

5 3

(B)

4 3

(C)

3

5 4

(D)

7

Locus of intersection of two perpendicular tangents to the given hyperbola is 2

7  55 (A) (x – 3)2 +  y   = 2 4 

2

7  25 (B) (x – 3)2 +  y   = 2 4 

2

7  7 (C) (x – 3) +  y   = 2 4   2

5.36

(D) none of these

 7 If origin is shifted to point  3,  and the axes are rotated through an angle θ in clockwise sense so  2

that equation of given hyperbola changes to the standard form 4 (A) tan–1   3

3 (B) tan–1   4

5 (C) tan–1   3

x2 a2

–

y2 b2

= 1, then θ is :

3 (D) tan–1   5

33

 JEE (ADVANCED) - RRB 

Comprehension # 2

For the hyperbola

x2 a2

–

y2 b2

= 1 the normal at P meets the transverse axis AA′ in G and the conjugate axis

BB′ in g and CF be perpendicular to the normal from the centre. 5.37

5.38

5.39

PF . PG = K CB2, then K =

1 2

(A) 2

(B) 1

(C)

PF . Pg equals to PF . Pg = (A) CA2

(B) CF2

(C) CB2

(D) 4

(D) CA . CB

Locus of middle point of G and g is a hyperbola of eccentricity (A)

1 e 1 2

(B)

e e 1 2

(C) 2 e 2  1

(D)

e 2

Comprehension # 3 If a circle with centre C(α,β) intersects a rectangular hyperbola with centre L(h,k) at four points P(x1, y1), Q(x2,y2), R(x3, y3) and S(x4, y4), then the mean of the four points P, Q, R, S is the mean of the points C and L. In other words, the mid-point of CL coincides with the mean point of P,Q,R,S. Analytically, x1  x 2  x 3  x 4   h y1  y 2  y 3  y 4   k   and 4 2 4 2

5.40.

If four points are taken on the circle x2 + y2 = a2. A rectangular hyperbola (H) passes through these four points. If the centroid of the quadrilateral formed from these four points lie on the straight line 3x – 4y + 1 = 0 then find the locus of the centre of rectangular hyperbola (H). (A) 3x – 4y + 2 = 0 (B) 3x – 4y + 3 = 0 (C) 3x – 4y + 4 = 0 (D) None of these

5.41

A, B, C, D are the points of intersection of a circle and a rectangular hyperbola which have different centres. If AB passes through the centre of the hyperbola, then CD passes through : (A) centre of the hyperbola (B) centre of the circle (C) mid-point of the centres of circle and hyperbola (D) none of the points mentioned in the three options.

5.42

If the normals drawn at four concylic points on a rectangular hyperbola xy = c2 meet at point (α,β) then the centre of the circle has the coordinates (A) (α,β)

(B) (2α , 2β)

  (C)  ,   2 2

  (D)  ,   4 4

34

 JEE (ADVANCED) - RRB 

SECTION - V : MATRIX - MATCH TYPE 5.43

Match the following : Column – Ι

Column – ΙΙ

(A)

(p)

12

(q)

6

(r)

24

(s)

32

(t)

3

The area of the triangle that a tangent at a point of the hyperbola

(B)

x2 y2 – = 1 makes with its asymptotes is 16 9

If the line y = 3x + λ touches the curve 9x2 – 5y2 = 45, then |λ| is

(C)

If the chord x cos α + y sin α = p of the hyperbola x2 y2 – = 1 subtends a right angle at the centre, then 16 18

the diameter of the circle, concentric with the hyperbola, to which the given chord is a tangent is (D)

If λ be the length of the latus rectum of the hyperbola 2

2

16x – 9y + 32x + 36y – 164 = 0, then 3λ is equal to

5.44

Match the following : Column – Ι

(A)

Column – ΙΙ

A tangent drawn to hyperbola

x2 a

2

–

y2 b

2

 = 1 at P   6

(p)

17

(q)

32

(r)

16

(s)

24

(t)

8

forms a triangle of area 3a2 square units, with coordinate axes, then the square of its eccentricity is equal to (B)

If the eccentricity of the hyperbola x2 – y2 sec2θ = 5

3 times the eccentricity of the ellipse x2sec2θ + y 2= 25 then smallest positive value of θ is

(C)

For the hyperbola

asymptotes is (D)

6 , value of ‘p’ is p

x2 – y2 = 3, acute angle between its 3

 , then value of ‘’ is 24

For the hyperbola xy = 8 any tangent of it at P meets co-ordinate axes at Q and R then area of triangle CQR where ‘C’ is centre of the hyperbola is

35

 JEE (ADVANCED) - RRB 

5.45

Match the following :

(A)

Column – Ι

Column – ΙΙ

Value of c for which 3x2 – 5xy – 2y2 + 5x + 11y + c = 0

(p)

3

(q)

–4

(r)

–12

(s)

4

(t)

–6

are the asymptotes of the hyperbola 3x2 – 5xy – 2y2 + 5x + 11y – 8 = 0 (B)

If locus of a point, whose chord of contact with respect to the circle x2 + y2 = 4 is a tangent to the hyperbola xy = 1 is xy = c2 , then value of c2 is

(C)

If equation of a hyperbola whose conjugate axis is 5 and distance 2

2

between its foci is 13, is ax – by = c where a and b are coprime natural numbers, then value of (D)

ab is c

If the vertex of a hyperbola bisects the distance between its centre and the corresponding focus, then ratio of square of its conjugate axis to the square of its transverse axis is

SECTION - VI : INTEGER ANSWER TYPE

5.46

2 2 Chords of the circle x2 + y2 = 4, touch the hyperbola x  y  1 . The locus of their middle points is the 4 16

curve (x2 + y2)2 = λx2 – 16y2, then find λ

5.47

If a variable line has its intercepts on the co-ordinates axes e, e′, where

e e , are the eccentricities 2 2

of a hyperbola and its conjugate hyperbola, then this line always touches the circle x2 + y2 = r2, where r=

36