Anal

ANALYTIC GEOMETRY Problem 1: ECE Board April 1999 The linear distance between –4 and 17 on the number line is A. 13 B. 21 C. –17 D. –13 Problem 2: EE Board April 1994 Find the distance between A (4, –3) and B (–2, 5).    

A. 11 B. 9 C. 10 D.8 Problem 3: If the distance between points (3, y) and (8, 7) is 13, then y is equal to    

A. 5 or –5 B. 5 or 19 C. 19 D. –5 or 19 Problem 4: Find the coordinate of a point equidistant from (1, -6), (5, -6) and (6, -1).    

A. (2, -2) B. (3, -2) C. (3, -3) D. (2, -3) Problem 5: EE Board April 1995 The line segment connecting (x, 6) and (9, y) is bisected by the point (7, 3). Find the values of x and y.    

A. 14, 6 B. 33, 12 C. 5, 0 D. 14, 6 Problem 6: If (-2, -4) is the midpoint of (6, -7) and (x, y), then the values of x and y are    

  

A. x = 2, y = 1 B. x = -10, y = -1 C. x = 10, y = -1

 D. x = -8, y = -1 Problem 7: ECE Board November 1998 Determine the coordinates of the point which is three-fifths of the way from the point (2, -5) to the point (-3, 5).

A. (-1, 1) B. (-2, -1) C. (-1, -2) D. (1, -1) Problem 8: ECE Board April 1998 The segment from (-1, 4) to (2, -2) is extended three times its own length. The terminal point is    

A. (11, -24) B. (-11, -20) C. (11, -18) D. (11, -20) Problem 9: The points (a, 1), (b, 2) and (c, 3) are collinear. Which of the following is true?    

A. c – b = c – a B. c – b = b – a C. c – a = a – b D. c – a = b – a Problem 10: If the slope of the line connecting the origin and point P is ¾, find the abscissa of P if its ordinate is 6.    

A. 2 B. 6 C. 7 D. 8 Problem 11: ECE Board April 1999 Find the inclination of the line passing through (-5, 3) and (10, 7).    

A. 14.73 B. 14.93 C. 14.83 D. 14.63 Problem 12: Find the angle formed by the lines 2x + y – 8 = 0 and x + 3y + 4 = 0.    



A. 30º

B. 35º C. 45º D. 60º Problem 13: Find the angle between the lines 3x + 2y = 6 and x + y = 6.   

A. 12º 20’ B. 11º 19’ C. 14º 25’ D. 13º 06’ Problem 14: What is the acute angle between the lines y = 3x + 2 and y = 4x + 9?    

A. 4.4º B. 28.3º C. 5.2º D. 18.6º Problem 15: EE Board October 1997 Find the distance of the line 3x + 4y = 5 from the origin.    

A. 4 B. 3 C. 2 D. 1 Problem 16: CE Board November 1992 The two points on the lines 2x = 3y + 4 = 0 which are at a distance 2 from the line 3x + 4y – 6 = 0 are?    

A. (-5, 1) and (-5, 2) B. (64, -44) and (4, -4) C. (8, 8) and (12, 12) D. (44, -64) and (-4, 4) Problem 17: CE Board November 1992 The distance from the point (2, 1) to the line 4x – 3y + 5 = 0 is?    

A. 1 B. 2 C. 3 D. 4 Problem 18: CE Board November 1996 Determine the distance from (5, 10) to the line x – y = 0.    

 

A. 3.33 B. 3.54

C. 4.23 D. 5.45 Problem 19: The distance from a point (1, 3) to the line 4x + 3y + 12 = 0 is  

A. 4 units B. 5 units C. 6 units D. 7 units Problem 20: CE Board May 1992 Find the distance between the given lines 4x – 3y = 12 and 4x – 3y = -8.    

A. 3 B. 4 C. 5 D. 6 Problem 21: EE Board April 1995 Find the distance between the lines, 3x + y – 12 = 0 and 3x + y – 4 = 0.    

Problem 22: ME Board October 1996 What is the length of the line with a slope of 4/3 from a point (6, 4) to the yaxis? A. 10 B. 25 C. 50 D. 75 Problem 23: ME Board April 1998 Find the slope of the line defined by y – x = 5.    

A. 1 B. 1/4 C. -1/2 D. 5 + x Problem 24: CE Board November 1995    

What is the slope of the line 3x + 2y + 1 = 0? A. 3/2 B. 2/3 C. -3/2 D. -2/3 Problem 25: ECE Board November 1990 In a Cartesian coordinates, the vertices of a triangle are defined by the following points: (-2, 0) and (3, 3). What is the area?    

A. 8 sq. units B. 9 sq. units C. 10 sq. units D. 11 sq. units Problem 26: EE Board April 1994 Given three vertices of a triangle whose coordinates are A (1, 1), B (3, -3) and (5, -3). Find the area of the triangle.    

A. 3 B. 4 C. 5 D. 6 Problem 27: ECE Board November 1990 In a Cartesian coordinates, the vertices of a square are: (1, 1), (0, 8), (4, 5) and (-3, 4). What is the area?    

A. 20 sq. units B. 30 sq. units C. 25 sq. units D. 35 sq. units Problem 28: EE Board April 1997 A line passes thru (1, -3) and (-4, 2. Write the equation of the line in slopeintercept form.    

A. y – 4 = x B. y = -x – 2 C. y = x – 4 D. y – 2 = x Problem 29: EE Board October 1997 What is the x-intercept of the line passing through (1, 4) and (4, 1)?    

  

A. 4.5 B. 5 C. 4

 D. 6 Problem 30: ME Board April 1997 Find the equation of the straight line with a slope of 3 and a y-intercept of 1.

A. 3x + y – 1 = 0 B. 3x – y + 1 = 0 C. x + 3y + 1 = 0 D. x – 3y – 1 = 0 Problem 31: ECE Board April 1999 If the points (-2, 3), (x, y) and (-3, 5) lie on a straight line, then the equation of the line is _______.    

A. x – 2y – 1 = 0 B. 2x + y – 1 = 0 C. x + 2y – 1 = 0 D. 2x + y + 1 = 0 Problem 32: ME Board April 1998 The equation of a line that intercepts the x-axis at x = 4 and the y-axis at y = -6 is,    

A. 3x + 2y = 12 B. 2x – 3y = 12 C. 3x – 2y = 12 D. 2x – 3y = 12 Problem 33: A line with an inclination of 45º passes through (-5/2, -9/2). What is the xcoordinate of a point on the line if its corresponding y-coordinate is 6?    

A. 6 B. 7 C. 8 D. 9 Problem 34: Find the equation of the line passing through the origin and with a slope of 6?    

A. y – 6x = 0 B. y = -6 C. x + y = -6 D. 6x + y = 0 Problem 35: Find the equation of the line if the x-intercept and y-intercept are -2 and 4, respectively.    

A. y – 2x – 4 = 0 B. y + 2x – 4 = 0 C. y – 2x + 4 = 0 D. y + 2x + 4 = 0 Problem 36: ECE Board April 1998 Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0.    

A. 5 B. 4 C. 3 D. 2 Problem 37: The line 2x – 3y + 2 = 0 is perpendicular to another line L1 of unknown equation. Find the slope of L1.    

A. 3/2 B. -3/2 C. 2/3 D. -2/3 Problem 38: A line through (-5, 2) and (1, -4) is perpendicular to the line through (x, -7) and (8, 7). Find the x.    

A. -4 B. -5 C. -6 D. -19/3 Problem 39: CE Board May 1996 What is the equation of the line that passes thru (4, 0) and is parallel to the line x – y – 2 = 0?    

A. x – y + 4 = 0 B. x + y + 4 = 0 C. x – y – 4 = 0 D. x – y = 0 Problem 40: Find the equation of the line through point (3, 1) and is perpendicular to the line x + 5y +5 = o.    

A. 5x – 2y = 14 B. 5x – y = 14 C. 2x – 5y = 14 D. 2x + 5y = 14 Problem 41:    

Find the equation of the perpendicular bisector of the line joining (5, 0) and (-7, 3). A. 8x + 2y + 11 = 0 B. 8x – 2y + 11 = 0 C. 8x – y + 11 = 0 D. 8x + y + 11 = 0 Problem 42: Which of the following lines is parallel to the line 3x – 2y + 6 = 0?    

A. 3x + 2y – 12 = 0 B. 4x – 9y = 6 C. 12x + 18y = 15 D. 15 x – 10y – 9 = 0 Problem 43: The equation of the line through (-3, -5) parallel to 7x + 2y – 4 = 0 is    

A. 7x + 2y + 31 = 0 B. 7x – 2y + 30 = 0 C. 7x – 2y – 4 = 0 D. 2x + 7y + 30 = 0 Problem 44: What is the equation of the line joining the points (3, -2) and (-7, 6)?    

A. 2x + 3y = 0 B. 4x – 5y = 22 C. 4x + 5y = 2 D. 5x + 4y = 7 Problem 45: What is the equation of the line passing through (-2, 6) with the x-intercept half the y-intercept?    

A. x – y =6 B. 2x + 2y + 2 = 0 C. 3x – y + 2 = 0 D. 2x + y – 2 = 0 Problem 46: CE Board May 1997 Find the slope of a line having a parametric equation of x = 2 + t and y = 5 – 3t.    

   

A. 2 B. 3 C. -2 D. -3

Problem 47: CE Board May 1998 Find the slope of the line having a parametric equation y = 4t + 6 and x = t + 1. A. 1 B. 2 C. 3 D. 4 Problem 48: ECE Board April 1999 Two vertices of a triangle are (2, 4) and (-2, 3) and the area is 2 square units, the locus of the third vertex is?    

A. 4x – y = 14 B. 4x + 4y = 14 C. x + 4y = 12 D. x – 4y = -14 Problem 49: ECE Board April 1998 Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axis.    

A. 3 B. 4 C. 5 D.2 Problem 50: ECE Board November 1998 A line passes through point (2, 2). Find the equation of the line if the length of the line segment intercepted by the coordinate axes is the square root of 5.    

A. 2x + y – 2 = 0 B. 2x – y – 2 = 0 C. 2x – y + 2 = 0 D. 2x + y + 2 = 0 51. State the quadrant in which the coordinate (15, -2) lies.    

A. I B. IV C. II D. III 52. Of what quadrant is A, if sec A is positive and csc A is negative?    

  

A. III B. I C. IV

 D. II 53. The segment from (-1, 4) to (2, -2) is extended three times its own length. The terminal point is

A. (11, -18) B. (11, -24) C. (11, -20) D. (-11, -20) 54. The midpoint of the line segment between P1(x, y) and P2(-2, 4) is Pm(2, -1). Find the coordinate of P1.    

A. (6, -5) B. (5, -6) C. (6, -6) D. (-6, 6) 55. Find the coordinates of the point P(2,4) with respect to the translated axis with origin at (1,3).    

A. (1, -1) B. (1, 1) C. (-1, -1) D. (-1, 1) 56. Find the median through (-2, -5) of the triangle whose vertices are (-6, 2), (2, -2), and (-2, -5).    

A. 3 B. 4 C. 5 D. 6 57. Find the centroid of a triangle whose vertices are (2, 3), (-4, 6) and (2, 6).    

A. (0, 1) B. (0, -1) C. (1, 0) D. (-1, 0) 58. Find the area of triangle whose vertices are A (-3, -1), B(5, 3) and (2, 8)    

A. 34 B. 36 C. 38 D. 32 59. Find the distance between the points (4, -2) and (-5, 1)    

A. 4.897 B. 8.947 C. 7.149 D. 9.487 60. Find the distance between A(4, -3) and B(-2, 5).    

A. 11 B. 8 C. 9 D. 10 61. If the distance between the points (8, 7) and (3, y) is 13, what is the value of y?    

A. 5 B. -19 C. 19 or -5 D. 5 or -19 62. The distance between the points (sin x, cos x) and (cos x, -sin x) is:    

A. 1 B. √2 C. 2 sin x cos x D. 4 sin x cos x 63. Find the distance from the point (2, 3) to the line 3x + 4y + 9 = 0.    

A. 5 B. 5.4 C. 5.8 D. 6.2 64. Find the distance from the point (5, -3) to the line 7x – 4y – 28 = 0.    

A. 2.62 B. 2.36 C. 2.48 D. 2.54 65. How far is the line 3x – 4y + 15 = 0 from the origin?    

A. 1 B. 2 C. 3 D. 4 66. Determine the distance from (5, 10) to the line x – y = 0    



A. 3.86

B. 3.54 C. 3.68 D. 3.72 67. The two points on the lines 2x + 3y +4 = 0 which are at distance 2 from the line 3x + 4y – 6 = 0 are:   

A. (-8, -8) and (-16, -16) B. (-44, 64) and (-5, 2) C. (-5.5, 1) and (-5, 2) D. (64, -44) and (4, -4) 68. The intercept form for algebraic straight-line equation is:    

A. (a/x) + (y/b) = 1 B. y = mx + b C. Ax + By + C = 0 D. (x/a) + (y/b) = 1 69. Find the slope of the line defined by y – x = 5    

A. 1 B. -1/2 C. ¼ D. 5 + x 70. The slope of the line 3x + 2y + 5 = 0 is:    

A. -2/3 B. -3/2 C. 3/2 D. 2/3 71. Find the slope of the line whose parametric equation is y = 5 – 3t and x = 2 + t.    

A. 3 B. -3 C. 2 D. -2 72. Find the slope of the curve whose parametric equations are    

x = -1 + t y = 2t   

A. 2 B. 3 C. 1

 D. 4 73. Find the angle that the line 2y – 9x – 18 = 0 makes with the x-axis.

A. 74.77° B. 4.5° C. 47.77° D. 77.47° 74. Which of the following is perpendicular to the line x/3 + y/4 = 1?    

A. x – 4y – 8 = 0 B. 4x – 3y – 6 = 0 C. 3x – 4y – 5 = 0 D. 4x + 3y – 11 = 0 75. Find the equation of the bisector of the obtuse angle between the lines 2x + y = 4 and 4x – 2y = 7    

A. 4y = 1 B. 8x = 15 C. 2y = 3 D. 8x + 4y = 6 76. The equation of the line through (1, 2) and parallel to the line 3x – 2y + 4 = 0 is:    

A. 3x – 2y + 1 = 0 B. 3x – 2y – 1 = 0 C. 3x + 2y + 1 = 0 D. 3x + 2y – 1 = 0 77. If the points (-3, -5), (x, y), and (3, 4) lie on a straight line, which of the following is correct?    

A. 3x + 2y – 1 = 0 B. 2x + 3y + 1 = 0 C. 2x + 3y – 1 = 0 D. 3x – 2y – 1 = 0 78. One line passes through the points (1, 9) and (2, 6), another line passes through (3, 3) and (-1, 5). The acute angle between the two lines is:    

A. 30° B. 45° C. 60° D. 135° 79. The two straight lines 4x – y + 3 = 0 and 8x – 2y + 6 = 0    



A. Intersects at the origin

B. Are coincident C. Are parallel D. Are perpendicular 80. A line which passes through (5, 6) and (-3. -4) has an equation of   

A. 5x + 4y + 1 = 0 B. 5x – 4y – 1 = 0 C. 5x – 4y + 1 = 0 D. 5x + y – 1 = 0 81. Find the equation of the line with slope of 2 and y-intercept of -3.    

A. y = -3x + 2 B. y = 2x – 3 C. y = 2/3 x + 1 D. y = 3x – 2 82. What is the equation of the line that passes through (4, 0) and is parallel to the line x – y – 2 = 0?    

A. y + x + 4 = 0 B. y – x + 4 = 0 C. y – x – 4 = 0 D. y + x – 4 = 0 83. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0    

A. 2 B. 3 C. 4 D. 5 84. The equation of a line that intercepts the x-axis at x = 4 and the y-axis at y = -6 is:    

A. 2x – 3y = 12 B. 3x + 2y = 12 C. 3x – 2y = 12 D. 2x – 37 = 12 85. How far from the y-axis is the center of the curve 2×2 + 2y2 + 10x – 6y – 55 = 0?    

   

A. -3.0 B. 2.75 C. -3.25 D. 2.5

86. Find the area of the circle whose center is at (2,-5) and tangent to the line 4x + 3y – 8 = 0. A. 6π B. 9π C. 3π D. 12π 87. Determine the area enclosed by the curve x2 – 10x + 4y + y2 = 196    

A. 15π B. 225π C. 12π D. 144π 88. Find the shortest distance from the point (1, 2) to appoint on the circumference of the circle defined by the equation x2 + y2 + 10x + 6y + 30 = 0.    

A. 5.61 B. 5.71 C. 5.81 D. 5.91 89. Determine the length of the chord common to the circles x2 + y2 = 64 and x2 + y2 – 16x – 0.    

A. 13.86 B. 12.82 C. 13.25 D. 12.28 90. If (3, -2) is on a circle with center (-1, 1), then the area of the circle is:    

A. 5π B. 25π C. 4π D. 3π 91. The radius of the circle 2×2 + 2y2 – 3x + 4y – 1 = 0 is:    

A. (√33)/4 B. 33/16 C. (√33)/3 D. 17 92. What is the radius of the circle with the following equation?    

x2 – 6x + y2 – 4y – 12 = 0

A. 3.46 B. 5 C. 7 D. 6 93. The diameter of a circle described by 9×2 + 9y2 = 16 is:    

A. 16/9 B. 4/3 C. 4 D. 8/3 94. Find the center of the circle x2 + y2 – 6x + 4y – 23 = 0.    

A. (3, -2) B. (3, 2) C. (-3, 2) D. (-3, -2) 95. Determine the equation of the circle whose center is at (4, 5) and tangent to the circle whose equation is x2 + y2 + 4x + 6y – 23 = 0.    

A. x2 + y2 – 8x + 10y – 25 = 0 B. x2 + y2 + 8x – 10y + 25 = 0 C. x2 + y2 – 8x – 10y + 25 = 0 D. x2 + y2 – 8x – 10y – 25 = 0 96. The equation of the circle with center at (-2, 3) and which is tangent to the line 20x – 21y – 42 = 0.    

A. x2 + y2 + 4x – 6y – 12 = 0 B. x2 + y2 + 4x – 6y + 12 = 0 C. x2 + y2 + 4x + 6y – 12 = 0 D. x2 + y2 – 4x – 6y – 12 = 0 97. A circle has a diameter whose ends are at (-3, 2) and (12, -6). Its Equation is:    

A. 4×2 + 4y2 – 36x + 16y + 192 = 0 B. 4×2 + 4y2 – 36x + 16y – 192 = 0 C. 4×2 + 4y2 – 36x – 16y – 192 = 0 D. 4×2 + 4y2 – 36x – 16y + 192 = 0 98. Find the equation of the circle with center on x + y = 4 and 5x + 2y + 1 = 0 and having a radius of 3.    

   

A. x2 + y2 + 6x – 16y + 64 = 0 B. x2 + y2 + 8x – 14y + 25 = 0 C. x2 + y2 + 6x – 14y + 49 = 0 D. x2 + y2 + 6x – 14y + 36 = 0

99. If (3, -2) lies on the circle with center (-1, 1) then the equation of the circle is: A. x2 + y2 + 2x – 2y – 23 = 0 B. x2 + y2 + 4x – 2y – 21 = 0 C. x2 + y2 + 2x – y – 33 = 0 D. x2 + y2 + 4x – 2y – 27 = 0 100. Find the equation of k for which the equation x2 + y2 + 4x – 2y – k = 0 represents a point circle.    

   

A. 5 B. -5 C. 6 D. -6  Below are the answers key for the Multiple Choice Questions in Analytic Geometry: Points, Lines and Circles – MCQs Part 1.  1. B. 21  Review: Solution for Number 1  2. C. 10  Review: Solution for Number 2  3. D. –5 or 19  Review: Solution for Number 3  4. C. (3, -3)  Review: Solution for Number 4  5. C. 5, 0  Review: Solution for Number 5  6. B. x = -10, y = –1  Review: Solution for Number 6  7. A. (-1, 1)  Review: Solution for Number 7  8. D. (11, -20)  Review: Solution for Number 8  9. B. c – b = b – a  Review: Solution for Number 9  10. D. 8  Review: Solution for Number 10  11. B. 14.93  Review: Solution for Number 11  12. C. 45º  Review: Solution for Number 12  13. B. 11º 19’  Review: Solution for Number 13  14. A. 4.4º  Review: Solution for Number 14

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15. D. 1 Review: Solution for Number 16. B. (64, -44) and (4, -4) Review: Solution for Number 17. B. 2 Review: Solution for Number 18. B. 3.54 Review: Solution for Number 19. B. 5 units Review: Solution for Number 20. B. 4 Review: Solution for Number 21. D Review: Solution for Number 22. A. 10 Review: Solution for Number 23. A. 1 Review: Solution for Number 24. C. -3/2 Review: Solution for Number 25. B. 9 sq. units Review: Solution for Number 26. B. 4 Review: Solution for Number 27. C. 25 sq. units Review: Solution for Number 28. B. y = -x – 2 Review: Solution for Number 29. B. 5 Review: Solution for Number 30. B. 3x – y + 1 = 0 Review: Solution for Number 31. D. 2x + y + 1 = 0 Review: Solution for Number 32. C. 3x – 2y = 12 Review: Solution for Number 33. C. 8 Review: Solution for Number 34. A. y – 6x = 0 Review: Solution for Number 35. B. y + 2x – 4 = 0 Review: Solution for Number 36. C. 3 Review: Solution for Number

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

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37. B. -3/2 Review: Solution for Number 37 38. C. –6 Review: Solution for Number 38 39. C. x – y – 4 = 0 Review: Solution for Number 39 40. B. 5x – y = 14 Review: Solution for Number 40 41. B. 8x – 2y + 11 = 0 Review: Solution for Number 41 42. D. 15 x – 10y – 9 = 0 Review: Solution for Number 42 43. A. 7x + 2y + 31 = 0 Review: Solution for Number 43 44. C. 4x + 5y = 2 Review: Solution for Number 44 45. D. 2x + y – 2 = 0 Review: Solution for Number 45 46. D. –3 Review: Solution for Number 46 47. D. 4 Review: Solution for Number 47 48. D. x – 4y = –14 Review: Solution for Number 48 49. A. 3 Review: Solution for Number 49 50. B. 2x – y – 2 = 0 Review: Solution for Number 50 Below are the answers key for the Multiple Choice Questions in Analytic Geometry: Points, Lines and Circles – MCQs Part 2. 51. B. IV 52. C. IV 53. C. (11, -20) 54. C. (6, -6) 55. B. (1, 1) 56. C. 5 57. A. (0, 1) 58. C. 38 59. D. 9.487 60. D. 10 61. C. 19 or –5 62. B. √2 63. B. 5.4 64. B. 2.36

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65. C. 3 66. B. 3.54 67. D. (64, -44) and (4, -4) 68. D. (x/a) + (y/b) = 1 69. A. 1 70. B. -3/2 71. B. –3 72. A. 2 73. D. 77.47° 74. C. 3x – 4y – 5 = 0 75. A. 4y = 1 76. A. 3x – 2y + 1 = 0 77. D. 3x – 2y – 1 = 0 78. B. 45° 79. B. Are coincident 80. B. 5x – 4y – 1 = 0 81. C. y = 2/3 x + 1 82. C. y – x – 4 = 0 83. C. 4 84. C. 3x – 2y = 12 85. D. 2.5 86. A. 6π 87. B. 225π 88. C. 5.81 89. A. 13.86 90. B. 25π 91. A. (√33)/4 92. B. 5 93. D. 8/3 94. A. (3, -2) 95. C. x2 + y2 – 8x – 10y + 25 = 0 96. A. x2 + y2 + 4x – 6y – 12 = 0 97. B. 4×2 + 4y2 – 36x + 16y – 192 = 0 98. C. x2 + y2 + 6x – 14y + 49 = 0 99. A. x2 + y2 + 2x – 2y – 23 = 0 100. B. -5