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AP ECET MATHEMATICS Multiple Choice Questions (MCQs)

AP ECET MATHEMATICS - Analytical Geometry MCQs

61. The standard equation of a circle with center at (h, k) and radius r is:

a) (x - h)² + (y - k)² = r²
b) x² + y² = r²
c) x² + y² + 2hx + 2ky = r²
d) (x + h)² + (y + k)² = r²

Answer: a) (x - h)² + (y - k)² = r²

62. The equation of a circle with ends of the diameter at points (x1, y1) and (x2, y2) is:

a) (x - (x1 + x2)/2)² + (y - (y1 + y2)/2)² = ((x2 - x1)² + (y2 - y1)²) / 4
b) (x - x1)² + (y - y1)² = (x2 - x1)² + (y2 - y1)²
c) (x + x1)² + (y + y1)² = (x2 - x1)² + (y2 - y1)²
d) (x - x2)² + (y - y2)² = ((x1 - x2)² + (y1 - y2)²) / 4

Answer: a) (x - (x1 + x2)/2)² + (y - (y1 + y2)/2)² = ((x2 - x1)² + (y2 - y1)²) / 4

63. The general equation of a circle is given by:

a) x² + y² + 2gx + 2fy + c = 0
b) x² + y² + gx + fy + c = 0
c) x² + y² + gxy + c = 0
d) x² + y² + 2gxy + c = 0

Answer: a) x² + y² + 2gx + 2fy + c = 0

64. The center of the circle given by the equation x² + y² + 4x - 6y - 12 = 0 is:

a) (-2, 3)
b) (2, -3)
c) (4, -6)
d) (-4, 6)

Answer: a) (-2, 3)

65. The radius of the circle with the equation x² + y² - 6x + 8y + 9 = 0 is:

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

Answer: b) 4

66. The standard equation of a parabola with vertex at the origin and focus at (0, f) is:

a) y² = 4fx
b) x² = 4fy
c) y² = 2fx
d) x² = 4f

Answer: a) y² = 4fx

67. For a parabola, the latus rectum is:

a) The distance between the focus and the directrix
b) The distance between the vertex and the focus
c) Perpendicular distance from the focus to the parabola
d) The length of the line segment joining the points where the parabola intersects the directrix

Answer: d) The length of the line segment joining the points where the parabola intersects the directrix

68. The equation of a hyperbola with transverse axis along the x-axis and center at the origin is:

a) (x²/a²) - (y²/b²) = 1
b) (y²/a²) - (x²/b²) = 1
c) (x²/b²) - (y²/a²) = 1
d) (x²/a²) + (y²/b²) = 1

Answer: a) (x²/a²) - (y²/b²) = 1

69. The foci of the hyperbola (x²/a²) - (y²/b²) = 1 are at:

a) (±√(a² + b²), 0)
b) (±√(a² - b²), 0)
c) (±√(b² - a²), 0)
d) (0, ±√(a² + b²))

Answer: b) (±√(a² - b²), 0)

70. The equation of a rectangular hyperbola is:

a) x² - y² = a²
b) x² + y² = a²
c) x² + y² = b²
d) x² - y² = b²

Answer: a) x² - y² = a²

71. The equation of a straight line passing through the origin and making an angle θ with the positive direction of the x-axis is:

a) y = tan(θ)x
b) y = cot(θ)x
c) y = sin(θ)x
d) y = cos(θ)x

Answer: a) y = tan(θ)x

72. The distance of the point (x1, y1) from the line Ax + By + C = 0 is given by:

a) |Ax1 + By1 + C| / √(A² + B²)
b) |Ax1 - By1 - C| / √(A² + B²)
c) |Ax1 + By1| / √(A² + B²)
d) |By1 + C| / √(A² + B²)

Answer: a) |Ax1 + By1 + C| / √(A² + B²)

73. The area of the triangle formed by the lines ax + by + c = 0, bx + cy + a = 0, and cx + ay + b = 0 is:

a) (abc) / 2
b) |(abc) / 2|
c) (a² + b² + c²) / 2
d) None of the above

Answer: b) |(abc) / 2|

74. The centroid of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is:

a) ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
b) ((x1 - x2 + x3)/3, (y1 + y2 - y3)/3)
c) ((x1 + x2 - x3)/3, (y1 - y2 + y3)/3)
d) (x1 + x2 + x3, y1 + y2 + y3)

Answer: a) ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

75. The length of the latus rectum of the ellipse x²/a² + y²/b² = 1 is:

a) 2a(1 - e²)
b) 2b(1 - e²)
c) 2a(1 + e²)
d) 2b(1 + e²)

Answer: b) 2b(1 - e²)

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AP ECET MATHEMATICS Multiple Choice Questions (MCQs)

AP ECET MATHEMATICS - Analytical Geometry MCQs

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