ACES REVIEW CENTER

REE ONLINE REVIEW

PRACTICE EXAM
ANALYTIC GEOMETRY
BY ENGR. JIMMY L. OCAMPO
0920 . 644 . 6246

INSTRUCTIONS: Shade the letter corresponding to the correct answer of your choice on the
answer sheet provided for. Absolutely no erasures.

1. Find the radius of the circle inscribed in the triangle determined by the lines y = x + 4, y = - x – 4
and y = 7x – 2.b
5 5 3 3
a. b. c. d.
√2 2√2 √2 2√2

2. Determine the eccentricity of the hyperbola xy = 8.c

a. 1.368 b. 1.521 c. 1.414 d. 1.732

3. The towers of a parabolic suspension bridge 300 m long are 60 m high and the lowest point of the
cable is 20 m above the roadway. Find the vertical distance from the roadway to the cable at 100 m
from the center.c
a. 17.78 m b. 12.88 m c. 37.78 m d. 32.86 m

1
4. What is the center and radius of a circle with an equation x2 + y2 – x – y = .d
64
√2
a. C (1, ½), R = 4 b. C (1/2, -½), R =
5
√33
c. C ( 1, 1), R= √5 d. C (1/2, ½), R =
8

5. Find the area of a quadrilateral having vertices at (2, -1), (4,3), (-1,2) and (-3,-2).c
a. 16 b. 17 c. 18 d. 14

6. Write the equation of a line with x-intercept a = 8 and y-intercept, b = -1.d

a. 8x + y – 8 = 0 b. x – 8y + 8 = 0 c. 8x – y + 8 = 0 d. x – 8y – 8 = 0

7. A line with equation y = mx + b passes through (-1/3, -6) and (2,1). Find the value of m.c
a. 1 b. 4 c. 3 d. 2

8. Find the area of the region inside the triangle with vertices (1,1), (3,2) and (2,4).a
a. 5/2 b. ½ c. 3/2 d. 7/2

9. The eccentricity of the hyperbola having the rectangular equation 3x^2 – 4y^2 – 24x + 16y + 20
= 0 is,c
a. 1.12 b. 1.22 c. 1.32 d. 1.42
10. Find the equation of the parabola whose vertex is the origin and whose focus is the point (0,2).b
a. x^2 = 16y b. x^2 = 8y c. x^2 = - 16y d. x^2 = 6y

11. What is the length of the latus rectum of the parabola x2 = -16y.c
a. 8 b. – 8 c. 16 d. – 16

12. Find the area of a triangle having vertices at -4 – i, 1 + 2i, 4 – 3i.c

a. 15 b. 16 c. 17 d. 18

13. If the line kx + 3y + 5 = 0 has a slope of 2/3, determine k.b

a. -3 b. -2 c. 3 d. 2

14. Find the eccentricity of a hyperbola whose transverse and conjugate axes are equal in length.c
a. 2 b. 3 c. sq. rt. of 2 d. sq. rt. of 3

15. The orbit of The Halley’s Comet around the sun is an ellipse with the sun as a focus and
eccentricity e = 0.967. The position where the comet is closest to the sun to the position where it is
farthest from the sun is approximately 3.365 x 10 raised to 9 miles. Approximately how closed the
comet gets to the sun in million miles.d
a. 28 b. 48 c. 13 d. 56

16. Find the length of the chord of the circle x^2 + y^2 + 4x + 6y – 32 = 0 if its distance from the
center of the circle is 5 m.b
a. 9. 84 b. 8.94 c. 7.89 d. 7.98

17. Find the length of the arc x^2 + y^2 = 25 from x = 2 to x = 4 in the first quadrant.c
a. 2.46 b. 1.83 c. 2.58 d. 1.96

18. For what value of the k will the line y = x + k be tangent to the hyperbola x^2 – 4y^2 – 48 = 0?b
a. +2, -2 b. +6, -6 c. +8, -8 d. +4, -4

19. Find the distance of the directrix of the ellipse from its center if the equation of the ellipse is 64x^2
+ 100 y^2 = 6400.c
a. 52/3 b. 55/3 c. 50/3 d. 56/3

20. Find the distance from the point (4,7) to the line (3x + 4y =1).a
a. 39/5 b. 36/5 c. 42/5 d. 46/5

21. A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus
is 30ft. If the distance across the top of the mirror is 64 in, how deep is the mirror of the center?a
a. 32/45 in b. 30 / 43 in c. 32/47 in d. 35/46 in

22. Find the slope of the line through the points (-2,5) and (7,1).d
a. 9/4 b. -9/4 c. 4/9 d. -4/9
23. For what values of k will the line kx + 5y = 2k have a y-intercept 4?d
a. 8 b. 7 c. 9 d. 10

24. Find the slope of the line defined by y – x = 5.a

a. 1 b. ¼ c. -1/2 d. 5
25. The radius of the circle x + y – 6x + 4y – 3 = 0 is,b
2 2

a. 3 b. 4 c. 5 d. 6

26. If the planes 5x – 6y – 7z = 0 and 3nx + 2y – mz + 1 = 0 are parallel, find m.d

a. -2/3 b. -4/3 c. -5/3 d. -7/3

27. If the equation of the directrix of the parabola is x – 5 = 0 and its focus is at (1,0), find the length
of its latus rectum.b
a. 6 b. 8 c. 10 d. 12

28. The line segment connecting (x, 6) and (9, y) is bisected by a point (7, 3). Find the value of x and
y.a
a. 5, 0 b. 4, 0 c. 5, 2 d. 4, 1

29. What is the height of the parabolic arch which has a span of 48 ft. and having a height of 20 ft. at
a distance of 16 ft. from the center of the span?c
a. 30 ft. b. 40 ft. c. 36 ft. d. 34 ft.

30. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0.b

a. 2 b. 3 c. 4 d. 5

31. The vertex of the parabola y2 – 2y + 6y + 3 = 0 is at,d

a. (-3, 3) b. (3, 3) c. (3, -3) d. (-3, -3)

32. Find the equation of the straight line which passes through the point (6, -3) and with an angle of
inclination of 45 degrees.d
a. x + y = 8 b. x - y = 8 c. x + y = 9 d. x - y = 9

33. If the distance between points A (2, 10, 4) and B (8, 3, z) is 9.434, what is the value of z?c
a. 4 b. 3 c. 6 d. 5

34. Find the rectangular coordinates of the point with polar coordinates r = 3 and theta = pi/6.b
a. (sq.rt of 3)/2, 2/3 b. 3(sq.rt of 3)/2, 3/2
c. 3(sq.rt of 2)/2, 2/3 d. sq.rt of 3, 3/2

35. Write the equation of the line with x intercept a = 4/5 and y intercept b = ½.b
a. 5x + 3y = 4 b. 5x + 8y = 4 c. 5x – 3y = 4 d. 10x + 9y = 8

36. The arch of an underpass is a semi-ellipse 60 ft wide and 20 ft high. Find the clearance at the edge
of a lane if the edge is 20 ft from the middle.a
a. 14.9 ft b. 12.8 ft c. 16.8 ft d. 18.2 ft
37. Express 9y = 4x + 36 in the intercept form.d
𝑥 𝑦 𝑥 𝑦 𝑥 𝑦 𝑥 𝑦
a. + = 1 b. + = - 1 c. + =1 d. + =1
9 4 −9 4 9 −4 −9 4

38. The coordinates of points A and B are A (-2, -3) and B (2, -5). What is an equation of the line that
is perpendicular to AB at its midpoint?d
a. 2y = -x – 8 b. 2y = x + 8 c. y = 2x + 4 d. y = 2x – 4

39. Find the area of a triangle whose vertices are (1,2), (-2,-1) and (3,0).b
a. 4 b. 6 c. 8 d. 10

2
40. Find the eccentricity of the ellipse when the length of its latus rectum is of the length of its major
3
axis.b
a. 0.56 b. 0.58 c. 0.62 d. 0.64

41. Find the rectangular coordinates of (0, 180°)a

a. (0,0) b. (1,0) c. (-1,0) d. (1,1)

42. Find the equation of a line parallel to y = 1 through (-1,1). a

a. y = 1 b. y = -1 c. y = 2 d. y = - 2

43. Find the equation of a line whose x – intercept a = 2, and y-intercept b = 3.b
𝑥 𝑦 𝑥 𝑦 𝑥 𝑦 𝑥 𝑦
a. + = 1 b. + = 1 c. - = 1 d. - = 1
3 2 2 3 3 2 3 2

1
44. Find the area enclosed by the curve r = 8sin2 𝜃 .c
2
a. 12π b. 6 π c. 24 π d. 20 π

45. A chord passing through the focus of the parabola y2 = 8x has one end at the point (8, 8). Where
is the other end of the court?d
a. (1/2, 2) b. (-1/2, -2) c. (-1/2, 2) d. (1/2, -2)

46. Find the radius of the circle inscribed in the triangle determined by the line y = x + 4, y = -x – 4
2
and y = x + 2.d
7
a. 2.29 b. 0.24 c. 1.57 d. 0.35

47. Write the equation of a line with x intercept a = -1, and y intercept b = 8.b
a. 8x + y – 8 = 0 b. 8x - y + 8 = 0 c. 8x + y + 8 = 0 d. 8x - y – 8 = 0

48. What is the distance between at any point P(x, y) on the ellipse b2x2 + a2y2 = a2b2 to its focus.d
a. by ± ax b. b ± ay c. ay ± bx d. a ± ex

49. Calculate the eccentricity of an ellipse whose major axis and latus rectum has lengths of 10 and
32/5, respectively.d
a. 0.3 b. 0.4 c. 0.8 d. 0.6
50. The equation of the directrix of the parabola y2 = 6x is,b
a. 2x – 3 = 0 b. 2x + 3 = 0 c. 3x – 2 = 0 d. 3x + 2 = 0

51. Find the area bounded by r = 4 (sqrt. of cos 2theta)b

a. 16 b. 8 c. 4 d. 12

52. What is the area of the ellipse whose eccentricity is 0.60 and whose major axis has a length of 8?a
a. 40.21 b. 41.20 c. 42.10 d. 40.12

53. If the line kx + 3y + 8 = 0 has a slope of 2/3, determine k.b

a. -3 b. -2 c. 3 d. 2

54. The vertex of the parabola y = (x – 1)2 + 2 is ___________.b

a. (-1, 2) b. (1, 2) c. (1, -2) d. (-1, -2)

55. What is the slope of the line through (-1, 2) and (4, -3)?b
a. 1 b. -1 c. 2 d. -2

56. The axis of the hyperbola through its foci is known as:c
a. Conjugate axis b. Major axis c. Transverse axis d. Minor axis

57. Classify the graph of the equation x^2 + xy + y^2 – 6 = 0,c

a. Circle b. Parabola c. Ellipse d. Hyperbola

58. A parabola having its axis along the x-axis passes through (-3, 6). Compute the length of latus
rectum if the vertex is at the origin.a
a. 12 b. 8 c. 6 d. 10

59. Find the volume of the tetrahedron bounded by the coordinate planes and the plane z = 6 – 2x +
3y.c
a. 4 b. 5 c. 6 d. 3

60. A line of slope 3 passes through the point (7, 10). If a point on the line has an abscissa 2, find its
ordinate.a
a. y = -5 b. y = 5 c. y = -6 d. y = 6

61. In the non-degenerate conic 3(x sq.) + 6xy + 5(y sq.) –x + y = 0. Classify what kind?c
a. Hyperbola b. Parabola c. Ellipse d. Circle

62. The inclination of the line determined by the points (4,0) and (5, sq. rt. of 3) is,c
a. 30° b. 45° c. 60° d. 65°

63. Given the ellipse (x squared / 25) + (y squared / 16) = 1, find the distance between foci.c
a. 4 b. 5 c. 6 d. 7
64. Find the area enclosed by r = 2 cos 3 theta.b
a. 2 b. pi c. 4 d. 2pi

65. Change y = x from rectangular to polar form.c

a. 𝜃 = 2π or 3𝜋⁄2 b. 𝜃 = 𝜋⁄3 or 4𝜋⁄3
c. 𝜃 = 𝜋⁄4 or 5𝜋⁄4 d. 𝜃 = 𝜋 or 3𝜋

66. Find the area bounded by the curve r2 = 4cos 2𝜃.c

a. 2 b. 2π c. 4 d. 4π

67. Find the equation of one of the medians of a triangle with vertices (0,0), (6,0) and (4,4).d
a. 2x – y = 10 b. 2x – 5y = 4 c. x + 10y = 4 d. x + 2y = 6

68. Find the equation of the bisector of the pair of acute angles formed by the line 4x + 2y = 9 and
2x – y = 8.d
a. y + 4x – 25 = 0 b. y + 8x – 25 = 0 c. y - 8x – 25 = 0 d. 8x – 25 = 0

69. Given the equation x2 – 2x + 3y2 + 6y = 0, find the length of the diameter whose slope is 1.b
a. 2√3 b. 2√2 c. 3√2 d. 3√3

70. Find the equation of the line, when the x-intercept a = -3, and y-intercept b = 4.c
𝑥 𝑦 𝑥 𝑦 𝑥 𝑦 𝑥 𝑦
a. + = 1 b. + = 1 c. + = 1 d. + = 1
4 −3 3 −4 −3 4 −4 3

71. Find the equation of the line that bisects the first and the third quadrant.b
a. x + y = 0 b. x – y = 0 c. 2x + y = 0 d. x - 2y = 0

72. An ellipse has its center at (0,0) with its axis horizontal. The distance between the vertices is 8
and its eccentricity is 0.5. Compute the length of the longest focal radius from point (2,3) on the
curve.c
a. 3 b. 4 c. 5 d. 6

73. The supporting cable of a suspension bridge hangs in the form of parabola from the top of 22m
tall towers which are 150m horizontally apart. If the lowest point on the cable is 7m above the
roadway, find the vertical distance in meters of the cable from the roadway at the point which is
15m from one of the supports.c
a. 9.6 b. 10.2 c. 16.6 d. 12.7

74. The angle from line L1, whose slope is -1/3to the line L2 is 135o. Find the slope of L2.c
a. 2 b.3 c. -2 d. -3

75. Determine the equation of the parabola with vertex at the origin, focus on the +x axis and with
its directrix 4 units from the focus.a
a. y^2 = 8x b. y^2 = 4x c. x^2 = 8y d. x^2 = 4y

76. Write the equation of a line with x-intercepts a = 8 and y-intercepts, b = -1.d
a. 8x+y-8=0 b. x-8y+8=0 c. 8x-y+8=0 d. x-8y-8=0
77. Find the area of the quadrilateral having ventricles at (2,-1), (4,3), (-1, 2) and(-3,-2).b
a. 16 b. 18 c. 17 d. 14

78. A parabolic cable has a span of 200 ft and a sag of 50 ft, the equation of the cable is,a
a. x^2 = 200y b. x^2 = 40y c. y^2 = 200x d. y^2 = 40x

79. Find the equation of the circle that passes through the vertex and the two ends of the latus
rectum of the parabola y^2 = 8x.b
a. x^2 + y^2 = 5x b. x^2 + y^2 = 10x
c. x^2 + y^2 = 8x d. x^2 + y^2 = 4x

80. The parabola y^2 = 4ax and the line x = p enclosed an area with the centroid at the focus of the
parabola. Find p in terms of a.b
a. 3/5a b. 5/3a c. 3/4a d. 2/5a

81. The latus rectum of the parabola 4px^2 = y is,d

a. p b. 2p c. -4p d. 4p

82. The vertex of the parabola y2-2x+6y+3=0 is at,d

a. (-3,3) b. (3,3) c. (3,-3) d. (-3,-3)

83. For what value of k will the line y = x+k be tangent to the hyperbola x^2-4y^2-48=0?b
a. +2,-2 b. +6, -6 c. +8, -8 d. +4, -4

84. A trough has an elliptical cross-section which is 18 inches wide on top and 12 inches deep. If the
water surface is 8in below the top, find the width of the water surface.c
a. 6 sq.rt of 2 b. 12 sq.rt of 2 c. 6 sq.rt of 5 d. 12 sq.rt of 5

85. The three points (1,-1, -3), (2,0, -1) and (a, b, 3) lie on straight line, find the values of a and b.a
a. 4,2 b. 2,4 c. 4,1 d. 4,3

86. Find the volume of a cube having its two faces laid in the planes 2x - y + 2z–3 = 0 and 6x-3y +
6z+ 8 = 0.d
a. 564/729 b. 546/729 c. 4319/729 d. 4913/729

87. Determine the radius of sphere whose equation is x2 + y2 + z2 - 2x + 8y + 16z + 65 = 0.a

a. 4 b. 5 c. 6 d. 7

88. Find the volume of the pyramid formed in the first octant by the plane 6x + 10y + 5z – 30 = 0
and the coordinate planes.d
a. 12 b. 13 c. 14 d. 15

89. Find the angle between the planes 3x – y +z - 5 = 0 and x + 2y + 2z + 2 = 0.d

a.62.45° b. 52.45° c. 82.45° d. 72.45°

90. Find the distance from the point (1,5,-3) to the plane 4x + y + 8z + 33 = 0.b
a. 1 b. 2 c. 3 d. 4
91. Find the equation of the asymptotes of the hyperbola (y – 5)2 – (x + 5)2 = 36.c
a. y = ± x + 5 b. y - 5 = ± (x - 5)
c. y – 5 = ± (x + 5) d. y + 5 = ± (x + 5)

92. Find the coordinates of the point where the segment joining the points (2,-2, 1) and (5, 1, -2)
crosses the xy plane.a
a. (3,-1,0) b. (3, 1, 0) c. (3, 0, 1) d. (0,3,-1)

93. Find the point on the line x = y = z that is equidistant from (3,0,5) and (1,-1,4).b
a. (1,1,1) b. (2,2,2) c. (3,3,3) d. (4,4,4)

94. Find the distance between lines x/1 = y/2 = z-6/3 and x/3 = y/2 = z/1.b
a. sq.rt of 5 b. sq.rt of 6 c. sq.rt of 7 d. sq.rt of 3

95. Find the equation of the parabola with vertical axis that passes through the point (0,2) and points
of intersection of the parabolas x^2 + 2x + 3y + 4 = 0 and x^2 – 3x + y + 3 = 0.b
a. x^2 – 8x – y – 3 = 0 b. x^2 – 8x – y + 2 = 0
c. y^2 – 8x – y – 3 = 0 d. y^2 – 8x – y + 2 = 0

96. Find the distance between parallel lines x/6 = y/-2 = z/1 and x-7/6 = y/-2 = z+1/1.c
a. 7 b. 5 c. 3 d. 4

97. Find the symmetric form equation of the line x – y + 2z = 2 and 2x + y – z = 1.a
a. x-1/-1 = y+1/5 = z/3 b. x+1/1 = y+1/5 = z/3
c. x-1/-1 = y-1/5 = z/3 d. x-1/-1 = y+1/5 = z/2

98. Find the point of intersection between the lines x/1 = y+3/2 = z+1/3 and x-3/2 = y/1 = z-1/-1.d
a. (1,1,2) b. (1,2,1) c. (2,1,1) d. (1,-1,2)

99. Find the parametric equation of the line passing through the point (1,7,2), parallel to the plane x
+ y + z = 2 and perpendicular to the line x = 2t + 1, y = (3t+5)/2 and z = (4t-1)/3.b
a. x = t – 1, y = 7 + 4t, z = 3t – 2 b. x = t + 1, y = 7 – 4t, z = 3t + 2
c. x = t + 1, y = 7 + 4t, z = 3t – 2 d. x = t + 1, y = 7 – 4t, z = 3t – 2

100. Find the direction of the line that is perpendicular to the lines whose directions are [3,2,-1] and
[1,-3,4] respectively.b
a. [5,13,11] b. [-5,13,11] c. [5,13,-11] d. [-5,-13,-11]