Analytical Geometry 2 Problem 1: CE Board May 1995 What is the radius of the circle x2 + y2 – 6y = 0?
A. 2
B. 3
C. 4
D.5 Problem 2: CE Board November 1995 What are the coordinates of the center of the curve x2 + y2 – 2x – 4y – 31 = 0?
A. (-1, -1)
B. (-2, -2)
C. (1, 2)
D. (2, 1) Problem 3: A circle whose equation is x2 + y2 + 4x +6y – 23 = 0 has its center at
A. (2, 3)
B. (3, 2)
C. (-3, 2)
D. (-2, -3) Problem 4: ME Board April 1998 What is the radius of a circle with the ff. equation: x2 – 6x + y2 – 4y – 12 = 0
A. 3.46
B. 7
C. 5
D.6 Problem 5: ECE Board April 1998 The diameter of a circle described by 9x2 + 9y2 = 16 is?
A. 4/3
B. 16/9
C. 8/3
D.4 Problem 6: CE Board May 1996 How far from the y-axis is the center of the curve 2x2 + 2y2 +10x – 6y – 55 = 0
A. -2.5
B. -3.0
C. -2.75
D.-3.25 Problem 7: What is the distance between the centers of the circles x2 + y2 + 2x + 4y – 3 = 0 and x2 + y2 + 2x – 8x – 6y + 7 = 0?
A. 7.07
B. 7.77
C. 8.07
D.7.87 Problem 8: CE Board November 1993 The shortest distance from A (3, 8) to the circle x2 + y2 + 4x – 6y = 12 is equal to?
A. 2.1
B. 2.3
C. 2.5
D.2.7 Problem 9: ME Board October 1996 The equation circle x2 + y2 – 4x + 2y – 20 = 0 describes:
A. A circle of radius 5 centered at the origin.
B. An eclipse centered at (2, -1).
C. A sphere centered at the origin.
D.A circle of radius 5 centered at (2, -1). Problem 10: EE Board April 1997 The center of a circle is at (1, 1) and one point on its circumference is (-1, -3). Find the other end of the diameter through (-1, -3).
A. (2, 4)
B. (3, 5)
C. (3, 6)
D. (1, 3) Problem 11: Find the area (in square units) of the circle whose equation is x2 + y2 = 6x – 8y.
A. 20 π
B. 22 π
C. 25 π
D. 27 π Problem 12: Determine the equation of the circle whose radius is 5, center on the line x = 2 and tangent to the line 3x – 4y + 11 = 0. A. (x – 2)2 + (y – 2)2 = 5 B. (x – 2)2 + (y + 2)2 = 25 C. (x – 2)2 + (y + 2)2 = 5 D. (x – 2)2 + (y – 2)2 = 25 Problem 13: Find the equation of the circle with the center at (-4, -5) and tangent to the line 2x + 7y – 10 = 0. A. x2 + y2 + 8x – 10y – 12 = 0 B. x2 + y2 + 8x – 10y + 12 = 0 C. x2 + y2 + 8x + 10y – 12 = 0 D. x2 + y2 – 8x + 10y + 12 = 0 Problem 14: ECE Board April 1998 Find the value of k for which the equation x2 + y2 + 4x – 2y – k = 0 represents a point circle.
A. 5
B. 6
C. -6
D. -5 Problem 15: ECE Board April 1999 3x2 + 2x – 5y + 7 = 0. Determine the curve.
A. Parabola
B. Ellipse
C. Circle
D. Hyperbola Problem 16: CE Board May 1993, CE Board November 1993, ECE Board April 1994 The focus of the parabola y2 = 16x is at
A. (4, 0)
B. (0, 4)
C. (3, 0)
D. (0, 3) Problem 17: CE Board November 1994 Where is the vertex of the parabola x2 = 4(y – 2)?
A. (2, 0)
B. (0, 2)
C. (3, 0)
D. (0, 3) Problem 18: ECE Board April 1994, ECE Board April 1999 Find the equation of the directrix of the parabola y2 = 16x.
A. x = 2
B. x = -2
C. x = 4
D. x = -4 Problem 19: Given the equation of a parabola 3x + 2y2 – 4y + 7 = 0. Locate its vertex.
A. (5/3, 1)
B. (5/3, -1)
C. -(5/3, -1)
D. (-5/3, 1) Problem 20: ME Board April 1997 In the equation y = - x2 + x + 1, where is the curve facing?
A. Upward
B. Facing left
C. Facing right
D. Downward Problem 21: CE Board May 1995 What is the length of the length of the latus rectum of the curve x2 = 20y?
Problem 22: EE Board October 1997
Find the location of the focus of the parabola y2 + 4x – 4y – 8 = 0.
A. (2.5, -2)
B. (3, 1)
C. (2, 2)
D. (-2.5, -2) Problem 23: ECE Board April 1998 Find the equation of the axis of symmetry of the function y = 2x2 – 7x + 5.
A. 7x + 4 = 0
B. 4x + 7 = 0
C. 4x – 7 = 0
D. x – 2 = 0 Problem 24: A parabola has its focus at (7, -4) and directrix y = 2. Find its equation. A. x2 + 12y – 14x + 61 = 0 B. x2 – 14y + 12x + 61 = 0 C. x2 – 12x + 14y + 61 = 0 D. none of the above Problem 25: A parabola has its axis parallel to the x-axis, vertex at (-1, 7) and one end of the latus rectum at (-15/4, 3/2). Find its equation. A. y2 – 11y + 11x – 60 = 0 B. y2 – 11y + 14x – 60 = 0 C. y2 – 14y + 11x + 60 = 0 D. none of the above Problem 26: ECE Board November 1997 Compute the focal length and the length of the latus rectum of the parabola y2 + 8x – 6y + 25 = 0.
A. 2, 8
B. 4, 16
C. 16, 64
D. 1, 4 Problem 27: Given a parabola (y – 2)2 = 8(x – 1). What is the equation of its directrix?
A. x = -3
B. x = 3
C. y = -3
D. y = 3 Problem 28: ME Board October 1997 The general equation of a conic section is given by the following equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. A curve maybe identified as an ellipse by which of the following conditions? A. B2 – 4AC < 0 B. B2 – 4AC = 0 C. B2 – 4AC > 0 D. B2 – 4AC = 1 Problem 29: CE Board November 1994 What is the area enclosed by the curve 9x2 + 25y2 – 225 = 0?
A. 47.1
B. 50.2
C. 63.8
D. 72.3 Problem 30: ECE Board April 1998 Point P (x, y) moves with a distance from point (0, 1) one-half of its distance from line y = 4. The equation of its locus is? A. 2x2 – 4y2 = 5 B. 4x2 + 3y2 = 12 C. 2x2 + 5y3 = 3 D. x2 + 2y2 = 4 Problem 31: The lengths of the major and minor axes of an ellipse are 10 m and 8 m, respectively. Find the distance between the foci.
A. 3
B. 4
C. 5
D. 6 Problem 32: The equation 25x2 + 16y2 – 150x + 128y + 81 = 0 has its center at?
A. (3, -4)
B. (3, 4)
C. (4, -3)
D. (3, 5) Problem 33: EE Board October 1997 Find the major axis of the ellipse x2 + 4y2 – 2x – 8y + 1 = 0.
A. 2
B. 10
C. 4
D. 6 Problem 34: CE Board May 1993 The length of the latus rectum for the ellipse is equal to?
A. 2
B. 3
C. 4
D. 5 Problem 35: An ellipse with an eccentricity of 0.65 and has one of its foci 2 units from the center. The length of the latus rectum is nearest to?
A. 3.5 units
B. 3.8 units
C. 4.2 units
D. 3.2 units Problem 36: An earth satellite has an apogee of 40,000 km and a perigee of 6,600 km. Assuming the radius of the earth as 6,400 km, what will be the eccentricity of the elliptical path described by the satellite with the center of the earth at one of the foci?
A. 0.46
B. 0.49
C. 0.52
D. 0.56
Problem 37: ECE Board April 1998 The major axis of the elliptical path in which the earth moves around the sun is approximately 168,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth.
A. 93,000,000 miles
B. 91,450,000 miles
C. 94,335,100 miles
D. 94,550,000 miles Problem 38: CE Board November 1992 The earth’s orbit is an ellipse with the sun at one of the foci. If the farthest distance of the sun from the earth is 105.5 million km and the nearest distance of the sun from the earth is 78.25 million km, find the eccentricity of the ellipse.
A. 0.15
B. 0.25
C. 0.35
D. 0.45 Problem 39: An ellipse with center at the origin has a length of major axis 20 units. If the distance from center of ellipse to its focus is 5, what is the equation of its directrix?
A. x = 18
B. x = 20
C. x = 15
D. x = 16 Problem 40: What is the length of the latus rectum of 4x2 + 9y2 + 8x – 32 = 0?
A. 2.5
B. 2.7
C. 2.3
D. 2.9 Problem 41: EE Board October 1993 4x2 – y2 = 16 is the equation of a/an?
A. parabola
B. hyperbola
C. circle
D. ellipse Problem 42: EE Board October 1993 Find the eccentricity of the curve 9x2 – 4y2 – 36x + 8y = 4.
A. 1.80
B. 1.92
C. 1.86
D. 1.76 Problem 43: CE Board November 1995 How far from the x-axis is the focus F of the hyperbola x2 – 2y2 + 4x + 4y + 4 = 0?
A. 4.5
B. 3.4
C. 2.7
D. 2.1 Problem 44: EE Board October 1994 The semi-transverse axis of the hyperbola is?
A. 2
B. 3
C. 4
D. 5 Problem 45: CE Board May 1996 What is the equation of the asymptote of the hyperbola?
A. 2x – 3y = 0
B. 3x – 2y = 0
C. 2x – y = 0
D. 2x + y = 0 Problem 46: EE Board April 1994 Find the equation of the hyperbola whose asymptotes are y = ± 2x and which passes through (5/2, 3). A. 4x2 + y2 + 16 = 0 B. 4x2 + y2 – 16 = 0 C. x2 – 4y2 – 16 = 0 D. 4x2 – y2 = 16 Problem 47: Find the equation of the hyperbola with vertices (-4, 2) and (0, 2) and foci (-5, 2) and (1, 2). A. 5x2 – 4y2 + 20x +16y – 16 = 0 B. 5x2 – 4y2 + 20x – 16y – 16 = 0 C. 5x2 – 4y2 – 20x +16y + 16 = 0 D. 5x2 + 4y2 – 20x +16y – 16 = 0 Problem 48: Find the distance between P1 (6, -2, -3) and P2 (5, 1, -4).
Problem 49: The point of intersection of the planes x + 5y – 2z = 9; 3x – 2y + z = 3 and x + y + z = 2 is at?
A. (2, 1, -1)
B. (2, 0, -1)
C. (-1, 1, -1)
D. (-1, 2, -1) Problem 50: ME Board April 1997 What is the radius of the sphere center at the origin that passes the point 8, 1, 6?
Problem 51: The equation of a sphere with center at (-3, 2, 4) and of radius 6 units is? A. x2 + y2 + z2 +6x – 4y – 8z = 36 B. x2 + y2 + z2 +6x – 4y – 8z = 7 C. x2 + y2 + z2 +6x – 4y + 8z = 6 D. x2 + y2 + z2 +6x – 4y + 8z = 36 Problem 52: EE Board April 1997 Find the polar question of the circle, if its center is at (4, 0) and the radius 4.
A. r – 8 cos θ = 0
B. r – 6 cos θ = 0
C. r – 12 cos θ = 0
D. r – 4 cos θ = 0 Problem 53: ME Board October 1996 What are the x and y coordinates of the focus of the iconic section described by the following equation? (Angle θ corresponds to a right triangle with adjacent side x, opposite side y and the hypotenuse r.) r sin2 θ = cos θ
A. (1/4, 0)
B. (0, π/2)
C. (0, 0)
D. (-1/2, 0) Problem 54: Find the polar equation of the circle of radius 3 units and center at (3, 0).
A. r = 3 cos θ
B. r = 3 sin θ
C. r = 6 cos θ
D. r = 9 sin θ Problem 55: EE Board October 1997
Given the polar equation r = 5 sin θ. Determine the rectangular coordinate (x, y) of a point in the curve when θ is 30º.
A. (2.17, 1.25)
B. (3.08, 1.5)
C. (2.51, 4.12)
D. (6, 3)
51. The vertex of the parabola y2 – 2x + 6y + 3 = 0 is at:
A. (-3, 3)
B. (3, 3)
C. (-3, 3)
D. (-3, -3) 52. The length of the latus rectum of the parabola y2 = 4px is:
A. 4p
B. 2p
C. P
D. -4p 53. Given the equation of the parabola: y2 – 8x – 4y – 20 = 0. The length of its latus rectum is:
A. 2
B. 4
C. 6
D. 8 54. What is the length of the latus rectum of the curve x2 = –12y?
A. 12
B. -3
C. 3
D. -12 55. Find the equation of the directrix of the parabola y2 = 6x.
A. x = 8
B. x = 4
C. x = -8
D. x = -4
56. The curve y = –x2 + x + 1 opens:
A. Upward
B. To the left
C. To the right
D. Downward 57. The parabola y = –x2 + x + 1 opens:
A. To the right
B. To the left
C. Upward
D. Downward 58. Find the equation of the axis of symmetry of the function y = 2x2 – 7x + 5.
A. 4x + 7 = 0
B. x – 2 = 0
C. 4x – 7 = 0
D. 7x + 4 = 0
59. Find the equation of the locus of the center of the circle which moves so that it is tangent to the y-axis and to the circle of radius one (1) with center at (2,0). A. x2 + y2 – 6x + 3 = 0 B. x2 – 6x + 3 = 0 C. 2x2 + y2 – 6x + 3 = 0 D. y2 – 6x + 3 = 0 60. Find the equation of the parabola with vertex at (4, 3) and focus at (4, -1). A. y2 – 8x + 16y – 32 = 0 B. y2 + 8x + 16y – 32 = 0 C. y2 + 8x – 16y + 32 = 0 D. x2 – 8x + 16y – 32 = 0 61. Find the area bounded by the curves x2 + 8y + 16 = 0, x – 4 = 0, the xaxis, and the y-axis.
A. 10.67 sq. units
B. 10.33 sq. units
C. 9.67 sq. units
D. 8 sq. units 62. Find the area (in sq. units) bounded by the parabolas x2 – 2y = 0 and x2 + 2y – 8 = 0
A. 11.7
B. 10.7
C. 9.7
D. 4.7 63. The length of the latus rectum of the curve (x – 2)2 / 4 = (y + 4)2 / 25 = 1 is:
A. 1.6
B. 2.3
C. 0.80
D. 1.52
64. Find the length of the latus rectum of the following ellipse: 25x2 + 9y2 – 300x –144y + 1251 = 0
A. 3.4
B. 3.2
C. 3.6
D. 3.0
65. If the length of the major and minor axes of an ellipse is 10 cm and 8 cm, respectively, what is the eccentricity of the ellipse?
A. 0.50
B. 0.60
C. 0.70
D. 0.80 66. The eccentricity of the ellipse x2/4 + y2 / 16 = 1 is:
A. 0.725
B. 0.256
C. 0.689
D. 0.866 67. An ellipse has the equation 16x2 + 9y2 + 32x – 128 = 0. Its eccentricity is:
A. 0.531
B. 0.66
C. 0.824
D. 0.93 68. The center of the ellipse 4x2 + y2 – 16x – 6y – 43 = 0 is at:
A. (2, 3)
B. (4, -6)
C. (1, 9)
D. (-2, -5)
69. Find the ratio of the major axis to the minor axis of the ellipse: 9x2 + 4y2 – 72x – 24y – 144 = 0
A. 0.67
B. 1.8
C. 1.5
D. 0.75 70. The area of the ellipse 9x2 + 25y2 – 36x – 189 = 0 is equal to:
A. 15π sq. units
B. 20π sq. units
C. 25π sq. units
D. 30π sq. units 71. The area of the ellipse is given as A = 3.1416 a b. Find the area of the ellipse 25x2 + 16y2 – 100x + 32y = 284
A. 86.2 square units
B. 62.8 square units
C. 68.2 square units
D. 82.6 square units
72. The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to the directrix is:
A. 6.532
B. 6.047
C. 0.6614
D. 6.222
73. Given an ellipse x2 / 36 + y2 / 32 = 1. Determine the distance between foci.
A. 2
B. 3
C. 4
D. 8 74. How far apart are the directrices of the curve 25x2 + 9y2 – 300x – 144y + 1251 = 0?
A. 12.5
B. 14.2
C. 13.2
D. 15.2
75. The major axis of the elliptical path in which the earth moves around the sun is approximately 186,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth.
A. 94,550,000 miles
B. 94,335.100 miles
C. 91,450,000 miles
D. 93,000,000 miles
76. Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and focus at (1, -1). A. 9x2 + 36y2 – 54x + 50y – 116 = 0 B. 4x2 + 25y2 + 54x – 50y – 122 = 0 C. 9x2 + 25y2 + 50x + 50y + 109 = 0 D. 9x2 + 25y2 + 54x + 50y – 119 = 0 77. Point P(x, y) moves with a distance from point (0, 1) one-half of its distance from line y = 4, the equation of its locus is A. 4x2 + 3y2 = 12 B. 2x2 - 4y2 = 5 C. x2 + 2y2 = 4 D. 2x2 + 5y3 = 3
78. The chords of the ellipse 64^2 + 25y^2 = 1600 having equal slopes of 1/5 are bisected by its diameter. Determine the equation of the diameter of the ellipse.
A. 5x – 64y = 0
B. 64x – 5y = 0
C. 5x +64y = 0
D. 64x + 5y = 0 79. Find the equation of the upward asymptote of the hyperbola whose equation is (x – 2)2 / 9 – (y + 4)2 / 16
A. 3x + 4y – 20 = 0
B. 4x – 3y – 20 = 0
C. 4x + 3y – 20 = 0
D. 3x – 4y – 20 = 0 80. The semi-conjugate axis of the hyperbola (x2/9) – (y2/4) = 1 is:
A. 2
B. -2
C. 3
D. -3 81. What is the equation of the asymptote of the hyperbola (x2/9) – (y2/4) = 1.
A. 2x – 3y = 0
B. 3x – 2y = 0
C. 2x – y = 0
D. 2x + y = 0
82. The graph y = (x – 1) / (x + 2) is not defined at:
A. 0
B. 2
C. -2
D. 1 83. The equation x2 + Bx + y2 + Cy + D = 0 is:
A. Hyperbola
B. Parabola
C. Ellipse
D. Circle 84. The general second degree equation has the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and describes an ellipse if: A. B2 – 4AC = 0 B. B2 – 4AC > 0 C. B2 – 4AC = 1 D. B2 – 4AC < 0 85. Find the equation of the tangent to the circle x2 + y2 – 34 = 0 through point (3, 5).
A. 3x + 5y -34 = 0
B. 3x – 5y – 34 = 0
C. 3x + 5y + 34 = 0
D. 3x – 5y + 34 = 0 86. Find the equation of the tangent to the curve x2 + y2 + 4x + 16y – 32 = 0 through (4, 0).
A. 3x – 4y + 12 = 0
B. 3x – 4y – 12 = 0
C. 3x + 4y + 12 = 0
D. 3x + 4y - 12 = 0 87. Find the equation of the normal to the curve y2 + 2x + 3y = 0 though point (-5,2)
A. 7x + 2y + 39 = 0
B. 7x - 2y + 39 = 0
C. 2x - 7y - 39 = 0
D. 2x + 7y - 39 = 0 88. Determine the equation of the line tangent to the graph y = 2x2 + 1, at the point (1, 3).
A. y = 4x + 1
B. y = 4x – 1
C. y = 2x – 1
D. y = 2x + 1 89. Find the equation of the tangent to the curve x2 + y2 = 41 through (5, 4).
A. 5x + 4y = 41
B. 4x – 5y = 41
C. 4x + 5y = 41
D. 5x – 4y = 41 90. Find the equation of a line normal to the curve x2 = 16y at (4, 1).
A. 2x – y – 9 = 0
B. 2x – y + 9 =
C. 2x + y – 9 = 0
D. 2x + y + 9 = 0 91. What is the equation of the tangent to the curve 9x2 + 25y2 – 225 = 0 at (0, 3)?
A. y + 3 = 0
B. x + 3 = 0
C. x – 3 = 0
D. y – 3 = 0 92. What is the equation of the normal to the curve x2 + y2 = 25 at (4, 3)?
A. 3x – 4y = 0
B. 5x + 3y = 0
C. 5x – 3y = 0
D. 3x + 4y = 0
93. The polar form of the equation 3x + 4y – 2 = 0 is:
A. 3r sin Ѳ + 4r cos Ѳ = 2
B. 3r cos Ѳ + 4r sin Ѳ = -2
C. 3r cos Ѳ + 4r sin Ѳ = 2
D. 3r sin Ѳ + 4r tan Ѳ = -2
94. The polar form of the equation 3x + 4y – 2 = 0 is: A. r2 = 8 B. r = Ѳ/(cos2 Ѳ + 2)
C. r = 8 D. r2 = 8/(cos2 Ѳ + 2)
95. the distance between points (5, 30°) and (-8, -50°) is:
A. 9.84
B. 10.14
C. 6.13
D. 12.14
96. Convert Ѳ = π/3 to Cartesian equation.
A. x = √3 x
B. y = x
C. 3y = √3 x
D. y =√3 x
97. The point of intersection of the planes x + 5y – 2z = 9, 3x – 2y + z = 3, and x + y + z = 2 is:
A. (2, 1, -1)
B. (2, 0, -1)
C. (-1, 1, -1)
D. (-1, 2, 1)
98. A warehouse roof needs a rectangular skylight with vertices (3, 0, 0), (3, 3, 0), (0, 3, 4), and (0, 0, 4). If the units are in meter, the area of the skylight is:
A. 12 sq. m.
B. 20 sq. m.
C. 15 sq. m.
D. 9 sq. m.
99. The distance between points in space coordinates are (3, 4, 5) and (4, 6, 7) is:
A. 1
B. 2
C. 3
D. 4
100. What is the radius of the sphere with center at origin and which passes through the point (8, 1, 6)?
A. 10
B. 9
R
C.√101
D. 10.5
