1. The equation of a circle centered at origin (0, 0) with radius r is:
A. x² + y² = r²
B. x² - y² = r²
C. (x - h)² + (y - k)² = r²
D. x + y = r
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Answer: A. x² + y² = r²
Explanation: Standard equation of a circle centered at origin is x² + y² = r².
2. The center of the circle (x - 3)² + (y + 4)² = 25 is:
A. (3, -4)
B. (-3, 4)
C. (3, 4)
D. (-3, -4)
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Answer: A. (3, -4)
Explanation: Comparing with (x - h)² + (y - k)² = r², h = 3 and k = -4.
3. What is the radius of the circle x² + y² - 4x - 8y - 5 = 0?
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Answer: A. 5
Explanation: g = -2, f = -4, c = -5. Radius r = √(g² + f² - c) = √(4 + 16 - (-5)) = √25 = 5.
4. Focus of parabola y² = 4ax is located at:
A. (a, 0)
B. (-a, 0)
C. (0, a)
D. (0, -a)
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Answer: A. (a, 0)
Explanation: Standard parabola y² = 4ax opens to the right with focus at (a, 0).
5. Directrix of parabola y² = -4ax is given by line:
A. x = a
B. x = -a
C. y = a
D. y = -a
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Answer: A. x = a
Explanation: Since focus is (-a, 0), the directrix is on opposite side: x = a.
6. Length of latus rectum of parabola x² = 4ay is:
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Answer: A. 4a
Explanation: Length of latus rectum for standard parabola is always 4a.
7. Find focus of parabola y² = 12x:
A. (3, 0)
B. (0, 3)
C. (-3, 0)
D. (6, 0)
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Answer: A. (3, 0)
Explanation: 4a = 12 ⇒ a = 3. Focus is (a, 0) = (3, 0).
8. The eccentricity 'e' of a circle is equal to:
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Answer: A. 0
Explanation: A circle is a special case of an ellipse with eccentricity e = 0.
9. The eccentricity 'e' of a parabola is equal to:
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Answer: A. 1
Explanation: Parabola has eccentricity e = 1 by definition (distance to focus equals distance to directrix).
10. The eccentricity 'e' of an ellipse satisfies:
A. 0 < e < 1
B. e = 1
C. e > 1
D. e = 0
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Answer: A. 0 < e < 1
Explanation: For an ellipse, eccentricity is always strictly between 0 and 1.
11. The eccentricity 'e' of a hyperbola satisfies:
A. e > 1
B. e = 1
C. 0 < e < 1
D. e = 0
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Answer: A. e > 1
Explanation: For any hyperbola, eccentricity e is strictly greater than 1.
12. Length of major axis of ellipse x²/a² + y²/b² = 1 (where a > b) is:
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Answer: A. 2a
Explanation: When a > b, major axis lies on x-axis and has total length 2a.
13. Length of minor axis of ellipse x²/a² + y²/b² = 1 (where a > b) is:
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Answer: A. 2b
Explanation: Minor axis lies along y-axis with total length 2b.
14. Relation between a, b, and c in an ellipse x²/a² + y²/b² = 1 (a > b) is:
A. c² = a² - b²
B. c² = a² + b²
C. a² = b² - c²
D. b² = a² + c²
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Answer: A. c² = a² - b²
Explanation: In an ellipse, semi-focal length c satisfies c² = a² - b².
15. Relation between a, b, and c in a hyperbola x²/a² - y²/b² = 1 is:
A. c² = a² + b²
B. c² = a² - b²
C. a² = b² + c²
D. b² = a² - c²
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Answer: A. c² = a² + b²
Explanation: In hyperbola, focal length is greater than transverse axis semi-length, c² = a² + b².
16. Length of latus rectum of ellipse x²/a² + y²/b² = 1 (a > b) is:
A. 2b²/a
B. 2a²/b
C. b²/a
D. a²/b
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Answer: A. 2b²/a
Explanation: Standard formula for latus rectum of ellipse (a > b) is 2b²/a.
17. Foci of hyperbola x²/a² - y²/b² = 1 are given by:
A. (±c, 0) where c = √(a² + b²)
B. (0, ±c) where c = √(a² + b²)
C. (±a, 0)
D. (0, ±b)
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Answer: A. (±c, 0) where c = √(a² + b²)
Explanation: Standard horizontal hyperbola has foci along x-axis at (±c, 0).
18. Foci of ellipse x²/25 + y²/9 = 1 are:
A. (±4, 0)
B. (0, ±4)
C. (±3, 0)
D. (±5, 0)
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Answer: A. (±4, 0)
Explanation: a² = 25, b² = 9 ⇒ c = √(25 - 9) = √16 = 4. Foci = (±4, 0).
19. Eccentricity of hyperbola x²/16 - y²/9 = 1 is:
A. 5/4
B. 4/5
C. 3/4
D. 5/3
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Answer: A. 5/4
Explanation: a² = 16, b² = 9 ⇒ c = √(16 + 9) = 5. Eccentricity e = c/a = 5/4.
20. Directrix of parabola x² = -8y is:
A. y = 2
B. y = -2
C. x = 2
D. x = -2
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Answer: A. y = 2
Explanation: 4a = 8 ⇒ a = 2. Parabola opens downwards, so directrix line is y = a ⇒ y = 2.
21. Length of transverse axis of hyperbola x²/9 - y²/16 = 1 is:
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Answer: A. 6
Explanation: a² = 9 ⇒ a = 3. Transverse axis length = 2a = 2(3) = 6.
22. Length of conjugate axis of hyperbola x²/9 - y²/16 = 1 is:
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Answer: A. 8
Explanation: b² = 16 ⇒ b = 4. Conjugate axis length = 2b = 2(4) = 8.
23. Equation of circle with center (0,0) passing through point (3, 4) is:
A. x² + y² = 25
B. x² + y² = 7
C. x² + y² = 12
D. x² + y² = 5
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Answer: A. x² + y² = 25
Explanation: Radius r = √(3² + 4²) = 5. Equation: x² + y² = r² = 25.
24. A hyperbola in which a = b is called a:
A. Rectangular hyperbola
B. Elliptic hyperbola
C. Conjugate hyperbola
D. Parabolic hyperbola
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Answer: A. Rectangular hyperbola
Explanation: When transverse axis equals conjugate axis (a = b), hyperbola is called rectangular hyperbola.
25. Eccentricity of a rectangular hyperbola is equal to:
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Answer: A. √2
Explanation: For rectangular hyperbola, a = b. e = √(1 + b²/a²) = √(1 + 1) = √2.
26. Vertices of ellipse x²/16 + y²/25 = 1 are:
A. (0, ±5)
B. (±4, 0)
C. (±5, 0)
D. (0, ±4)
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Answer: A. (0, ±5)
Explanation: Here b² = 25 > a² = 16, so major axis is along y-axis. Vertices are (0, ±b) = (0, ±5).
27. Length of latus rectum of hyperbola x²/a² - y²/b² = 1 is:
A. 2b²/a
B. 2a²/b
C. b²/a
D. a²/b
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Answer: A. 2b²/a
Explanation: Latus rectum formula for standard hyperbola x²/a² - y²/b² = 1 is 2b²/a.
28. Equation of parabola with vertex at (0,0) and focus at (0, -2) is:
A. x² = -8y
B. y² = -8x
C. x² = 8y
D. y² = 8x
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Answer: A. x² = -8y
Explanation: Focus on negative y-axis means form x² = -4ay. Here a = 2, so x² = -8y.
29. Area of circle x² + y² = 16 is:
A. 16π
B. 4π
C. 8π
D. 32π
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Answer: A. 16π
Explanation: r² = 16 ⇒ Area = πr² = 16π.
30. Position of point (2, 3) relative to circle x² + y² = 9 is:
A. Outside the circle
B. Inside the circle
C. On the circle
D. At center
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Answer: A. Outside the circle
Explanation: Substitute (2, 3): 2² + 3² = 4 + 9 = 13 > 9. Since value > 9, point lies outside.
