MCQ Mathematics Chapter 10: CONIC SECTIONS, HS 1st year

 

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Section: General Concepts

1. Who is credited with naming the curves parabola and hyperbola?

   • (a) Euclid
   • (b) Apollonius
   • (c) Descartes
   • (d) Newton
     Answer: (b) Apollonius

2. What is the general term used for curves like circles, ellipses, parabolas, and hyperbolas?

   • (a) Cartesian curves
   • (b) Conic sections
   • (c) Latus rectum
   • (d) Transverse sections
     Answer: (b) Conic sections

3. What determines the type of conic section formed by the intersection of a plane with a cone?

   • (a) The angle between the cone’s axis and the plane
   • (b) The diameter of the cone
   • (c) The rotation of the cone
   • (d) The material of the cone
     Answer: (a) The angle between the cone’s axis and the plane

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Section: Circles

4. The equation of a circle with center $(h, k)$ and radius $r$ is:

   • (a) $x^2 + y^2 = r^2$
   • (b) $(x - h)^2 + (y - k)^2 = r^2$
   • (c) $(x + h)^2 + (y + k)^2 = r^2$
   • (d) $(x - k)^2 + (y - h)^2 = r^2$
     Answer: (b) $(x - h)^2 + (y - k)^2 = r^2$

5. What is the radius of the circle given by the equation $x^2 + y^2 - 4x - 8y + 15 = 0$?

   • (a) $2$
   • (b) $3$
   • (c) $5$
   • (d) $7$
     Answer: (b) $3$

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Section: Parabolas

6. The equation of a parabola with its focus at $(a, 0)$ and directrix $x = -a$ is:

   • (a) $y^2 = 4ax$
   • (b) $x^2 = 4ay$
   • (c) $y^2 = -4ax$
   • (d) $x^2 = -4ay$
     Answer: (a) $y^2 = 4ax$

7. What is the length of the latus rectum for the parabola $y^2 = 16x$?

   • (a) 8
   • (b) 12
   • (c) 16
   • (d) 4
     Answer: (c) 16

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Section: Ellipses

8. If the equation of an ellipse is $\frac{x^2}{25} + \frac{y^2}{9} = 1$, what is the length of the major axis?

   • (a) 5
   • (b) 6
   • (c) 10
   • (d) 12
     Answer: (c) 10

9. What is the eccentricity ($e$) of an ellipse where $a = 5$ and $b = 4$?

   • (a) $e = \frac{3}{5}$
   • (b) $e = \frac{4}{5}$
   • (c) $e = \frac{1}{5}$
   • (d) $e = \frac{2}{5}$
     Answer: (a) $e = \frac{3}{5}$

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Section: Hyperbolas

10. The equation of a hyperbola with transverse axis along the x-axis is:

    • (a) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
    • (b) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
    • (c) $\frac{x^2}{b^2} - \frac{y^2}{a^2} = 1$
    • (d) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
      Answer: (b) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

11. Find the length of the latus rectum of a hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$:

• (a) $3$
• (b) $6$
• (c) $9$
• (d) $12$
  Answer: (d) $12$

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Circles

12. What is the equation of a circle with center $(-3, 2)$ and radius $4$?

    • (a) $(x + 3)^2 + (y - 2)^2 = 16$
    • (b) $(x - 3)^2 + (y + 2)^2 = 16$
    • (c) $(x - 3)^2 + (y - 2)^2 = 16$
    • (d) $(x + 3)^2 + (y + 2)^2 = 16$
      Answer: (a) $(x + 3)^2 + (y - 2)^2 = 16$

13. What is the radius of the circle given by the equation $(x - 4)^2 + (y + 5)^2 = 49$?

    • (a) 4
    • (b) 5
    • (c) 7
    • (d) 9
      Answer: (c) 7

14. If a circle passes through points $(3, 4)$ and $(5, 6)$ with center on $x + y = 2$, the radius of the circle is:

    • (a) 2
    • (b) $\sqrt{5}$
    • (c) 4
    • (d) $\sqrt{10}$
      Answer: (d) $\sqrt{10}$

15. What is the equation of a circle centered at the origin with radius $r$?

    • (a) $x^2 + y^2 = r$
    • (b) $x^2 + y^2 = r^2$
    • (c) $(x - h)^2 + (y - k)^2 = r$
    • (d) $(x - h)^2 + (y - k)^2 = r^2$
      Answer: (b) $x^2 + y^2 = r^2$

16. If the center of a circle is $(2, 3)$ and it passes through the origin, the radius is:

    • (a) $\sqrt{2}$
    • (b) $\sqrt{10}$
    • (c) $\sqrt{13}$
    • (d) $\sqrt{15}$
      Answer: (c) $\sqrt{13}$

17. Which of the following represents a degenerate conic section?

    • (a) Circle
    • (b) Ellipse
    • (c) Point
    • (d) Hyperbola
      Answer: (c) Point

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Parabolas

18. What is the axis of symmetry for the parabola $y^2 = 4ax$?

    • (a) x-axis
    • (b) y-axis
    • (c) line $y = x$
    • (d) line $x = y$
      Answer: (a) x-axis

19. The equation $x^2 = 12y$ represents a parabola that opens:

    • (a) Right
    • (b) Left
    • (c) Upward
    • (d) Downward
      Answer: (c) Upward

20. The directrix of the parabola $y^2 = 4x$ is given by:

    • (a) $x = 0$
    • (b) $x = -1$
    • (c) $x = -a$
    • (d) $x = -4$
      Answer: (c) $x = -a$

21. The latus rectum of the parabola $x^2 = -16y$ is:

    • (a) 4
    • (b) 8
    • (c) 16
    • (d) 12
      Answer: (b) 8

22. For the parabola $y^2 = 4ax$, the focus is located at:

    • (a) $(-a, 0)$
    • (b) $(a, 0)$
    • (c) $(0, a)$
    • (d) $(0, -a)$
      Answer: (b) $(a, 0)$

23. A parabola is symmetric about the y-axis when its equation is of the form:

    • (a) $y^2 = 4ax$
    • (b) $x^2 = 4ay$
    • (c) $y^2 = -4ax$
    • (d) $x^2 = -4ay$
      Answer: (b) $x^2 = 4ay$

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Ellipses

24. The sum of the distances from any point on an ellipse to its two foci is:

    • (a) Equal to $a + b$
    • (b) Equal to the length of the major axis
    • (c) Less than the length of the minor axis
    • (d) Equal to $c + a$
      Answer: (b) Equal to the length of the major axis

25. The foci of an ellipse are always located:

    • (a) On the minor axis
    • (b) On the major axis
    • (c) At the origin
    • (d) On the circumference
      Answer: (b) On the major axis

26. The eccentricity ($e$) of a circle is:

    • (a) 1
    • (b) 0
    • (c) Less than 1
    • (d) Greater than 1
      Answer: (b) 0

27. The equation of an ellipse with center at the origin and foci along the x-axis is:

    • (a) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
    • (b) $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$
    • (c) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
    • (d) $\frac{x^2}{b^2} - \frac{y^2}{a^2} = 1$
      Answer: (a) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

28. The length of the latus rectum for the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ is:

    • (a) $1.8$
    • (b) $4.5$
    • (c) $3.6$
    • (d) $9.2$
      Answer: (b) $4.5$

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Hyperbolas

29. The difference of the distances from any point on a hyperbola to its two foci is:

    • (a) Equal to the length of the transverse axis
    • (b) Equal to the length of the conjugate axis
    • (c) Equal to $a + b$
    • (d) Equal to $c - a$
      Answer: (a) Equal to the length of the transverse axis

30. The equation of a hyperbola with center at the origin and foci along the y-axis is:

    • (a) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
    • (b) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
    • (c) $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$
    • (d) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
      Answer: (b) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

31. The latus rectum of a hyperbola is a line segment:

    • (a) Parallel to the transverse axis
    • (b) Perpendicular to the transverse axis
    • (c) Along the conjugate axis
    • (d) Passing through the center
      Answer: (b) Perpendicular to the transverse axis

32. For the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$, the length of the transverse axis is:

    • (a) 4
    • (b) 8
    • (c) 10
    • (d) 12
      Answer: (b) 8

33. The eccentricity ($e$) of a hyperbola is always:

• (a) Less than 1
• (b) Equal to 1
• (c) Greater than 1
• (d) Undefined
  Answer: (c) Greater than 1

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Circles

34. The equation $(x - 3)^2 + (y + 4)^2 = 25$ represents a circle with center at:

    • (a) $(3, -4)$
    • (b) $(-3, 4)$
    • (c) $(-3, -4)$
    • (d) $(3, 4)$
      Answer: (a) $(3, -4)$

35. If the equation of a circle is $x^2 + y^2 - 6x + 8y + 9 = 0$, its center is located at:

    • (a) $(3, -4)$
    • (b) $(-3, 4)$
    • (c) $(-3, -4)$
    • (d) $(3, 4)$
      Answer: (a) $(3, -4)$

36. The radius of a circle passing through the origin and having center $(2, -3)$ is:

    • (a) 2
    • (b) $\sqrt{13}$
    • (c) $\sqrt{15}$
    • (d) $\sqrt{20}$
      Answer: (b) $\sqrt{13}$

37. The equation of a circle with diameter endpoints at $(1, 2)$ and $(3, 4)$ is:

    • (a) $(x - 2)^2 + (y - 3)^2 = 2$
    • (b) $(x - 2)^2 + (y - 3)^2 = 1$
    • (c) $(x - 2)^2 + (y - 3)^2 = 4$
    • (d) $(x - 1)^2 + (y - 2)^2 = 1$
      Answer: (c) $(x - 2)^2 + (y - 3)^2 = 4$

38. If a circle is centered at $(0, 0)$ and has a radius $r$, the area of the circle is:

    • (a) $2\pi r$
    • (b) $\pi r^2$
    • (c) $r^2$
    • (d) $\pi r$
      Answer: (b) $\pi r^2$

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Parabolas

39. A parabola has its vertex at the origin and its focus at $(0, 3)$. The equation of the parabola is:

    • (a) $x^2 = 4y$
    • (b) $x^2 = -4y$
    • (c) $y^2 = 4x$
    • (d) $y^2 = -4x$
      Answer: (a) $x^2 = 4y$

40. The parabola $y^2 = -8x$ opens:

    • (a) Left
    • (b) Right
    • (c) Upward
    • (d) Downward
      Answer: (a) Left

41. For the parabola $y^2 = 12x$, the length of the latus rectum is:

    • (a) 6
    • (b) 12
    • (c) 24
    • (d) 3
      Answer: (b) 12

42. The equation $x^2 = 16y$ represents a parabola with its vertex at the origin and its focus at:

    • (a) $(0, 4)$
    • (b) $(0, -4)$
    • (c) $(4, 0)$
    • (d) $(-4, 0)$
      Answer: (a) $(0, 4)$

43. The parabola $y^2 = 4x$ intersects the x-axis at:

    • (a) $(0, 0)$
    • (b) $(4, 0)$
    • (c) $(-4, 0)$
    • (d) $(0, 4)$
      Answer: (a) $(0, 0)$

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Ellipses

44. The equation of an ellipse with foci at $(\pm 4, 0)$ and vertices at $(\pm 5, 0)$ is:

    • (a) $\frac{x^2}{25} + \frac{y^2}{9} = 1$
    • (b) $\frac{x^2}{16} + \frac{y^2}{9} = 1$
    • (c) $\frac{x^2}{9} + \frac{y^2}{25} = 1$
    • (d) $\frac{x^2}{25} + \frac{y^2}{16} = 1$
      Answer: (d) $\frac{x^2}{25} + \frac{y^2}{16} = 1$

45. If the equation of an ellipse is $\frac{x^2}{4} + \frac{y^2}{9} = 1$, its semi-major axis lies along:

    • (a) The x-axis
    • (b) The y-axis
    • (c) The origin
    • (d) The line $y = x$
      Answer: (b) The y-axis

46. The foci of an ellipse $\frac{x^2}{49} + \frac{y^2}{36} = 1$ are located at:

    • (a) $(\pm 7, 0)$
    • (b) $(\pm 5, 0)$
    • (c) $(0, \pm 5)$
    • (d) $(0, \pm 7)$
      Answer: (a) $(\pm 5, 0)$

47. The latus rectum of an ellipse is always:

    • (a) Parallel to the major axis
    • (b) Perpendicular to the major axis
    • (c) Along the minor axis
    • (d) Along the origin
      Answer: (b) Perpendicular to the major axis

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Hyperbolas

48. The equation of a hyperbola with foci at $(\pm 5, 0)$ and vertices at $(\pm 3, 0)$ is:

    • (a) $\frac{x^2}{9} - \frac{y^2}{16} = 1$
    • (b) $\frac{x^2}{16} - \frac{y^2}{9} = 1$
    • (c) $\frac{y^2}{9} - \frac{x^2}{16} = 1$
    • (d) $\frac{y^2}{16} - \frac{x^2}{9} = 1$
      Answer: (a) $\frac{x^2}{9} - \frac{y^2}{16} = 1$

49. The asymptotes of the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$ are given by:

    • (a) $y = \pm \frac{5}{4}x$
    • (b) $y = \pm \frac{4}{5}x$
    • (c) $x = \pm \frac{5}{4}y$
    • (d) $x = \pm \frac{4}{5}y$
      Answer: (b) $y = \pm \frac{4}{5}x$

50. The eccentricity of a hyperbola with transverse axis $2a = 8$ and $b = 6$ is:

    • (a) $\sqrt{2}$
    • (b) $\sqrt{3}$
    • (c) $\sqrt{5}$
    • (d) $\sqrt{7}$
      Answer: (c) $\sqrt{5}$

51. The conjugate axis of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ has length:

    • (a) 2
    • (b) 4
    • (c) 6
    • (d) 8
      Answer: (b) 4

52. The latus rectum of a hyperbola with equation $\frac{x^2}{36} - \frac{y^2}{25} = 1$ is of length:

• (a) $2.5$
• (b) $5$
• (c) $10$
• (d) $15$
  Answer: (c) $10$

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General Concepts

53. Conic sections can be formed by the intersection of a plane with a:

    • (a) Right circular cone
    • (b) Cylinder
    • (c) Sphere
    • (d) Cube
      Answer: (a) Right circular cone

54. Which of the following is not a degenerate form of a conic section?

    • (a) A pair of intersecting lines
    • (b) A single point
    • (c) A straight line
    • (d) A hyperbola
      Answer: (d) A hyperbola

55. The line perpendicular to the axis of symmetry and passing through the vertex of a parabola is called the:

    • (a) Focus
    • (b) Directrix
    • (c) Latus rectum
    • (d) Axis of the parabola
      Answer: (b) Directrix

56. The distance between the two foci of an ellipse is given by:

    • (a) $2b$
    • (b) $2c$
    • (c) $c^2 - b^2$
    • (d) $a^2 - b^2$
      Answer: (b) $2c$

57. For which conic section is the eccentricity ($e$) always equal to 1?

    • (a) Ellipse
    • (b) Parabola
    • (c) Circle
    • (d) Hyperbola
      Answer: (b) Parabola

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Circles

58. If the center of a circle is $(h, k)$ and its radius is $r$, the equation of the circle is:

    • (a) $(x + h)^2 + (y + k)^2 = r^2$
    • (b) $(x - h)^2 + (y - k)^2 = r^2$
    • (c) $(x - h)^2 - (y - k)^2 = r^2$
    • (d) $(x + h)^2 - (y + k)^2 = r^2$
      Answer: (b) $(x - h)^2 + (y - k)^2 = r^2$

59. The circle with equation $x^2 + y^2 = 25$ has its center at:

    • (a) $(0, 0)$
    • (b) $(5, 0)$
    • (c) $(0, 5)$
    • (d) $(5, 5)$
      Answer: (a) $(0, 0)$

60. A circle passing through $(1, 2)$ and having its center at $(0, 0)$ has a radius of:

    • (a) $2$
    • (b) $1$
    • (c) $\sqrt{5}$
    • (d) $5$
      Answer: (c) $\sqrt{5}$

61. The diameter of a circle with radius $r = 7$ is:

    • (a) 7
    • (b) 14
    • (c) 21
    • (d) 28
      Answer: (b) 14

62. If a circle has its center at $(0, 0)$ and radius $r = 4$, its area is:

    • (a) $8\pi$
    • (b) $16\pi$
    • (c) $12\pi$
    • (d) $4\pi$
      Answer: (b) $16\pi$

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Parabolas

63. For the parabola $y^2 = 16x$, the distance of the focus from the vertex is:

    • (a) 2
    • (b) 4
    • (c) 8
    • (d) 16
      Answer: (b) 4

64. The vertex of the parabola $x^2 = -8y$ is located at:

    • (a) $(0, 0)$
    • (b) $(0, -8)$
    • (c) $(0, 8)$
    • (d) $(-8, 0)$
      Answer: (a) $(0, 0)$

65. The equation $x^2 = -4y$ represents a parabola that opens:

    • (a) Downward
    • (b) Upward
    • (c) Right
    • (d) Left
      Answer: (a) Downward

66. The latus rectum of a parabola is defined as:

    • (a) A line parallel to the directrix
    • (b) A line perpendicular to the axis of symmetry passing through the focus
    • (c) A line segment joining the vertex to the directrix
    • (d) A line parallel to the axis of symmetry
      Answer: (b) A line perpendicular to the axis of symmetry passing through the focus

67. If the parabola $y^2 = 4x$ is shifted so that its vertex is at $(1, 2)$, its new equation is:

    • (a) $(y - 2)^2 = 4(x - 1)$
    • (b) $(y + 2)^2 = 4(x + 1)$
    • (c) $(x - 2)^2 = 4(y - 1)$
    • (d) $(x - 1)^2 = 4(y - 2)$
      Answer: (a) $(y - 2)^2 = 4(x - 1)$

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Ellipses

68. If the equation of an ellipse is $\frac{x^2}{16} + \frac{y^2}{9} = 1$, the semi-major axis has a length of:

    • (a) 4
    • (b) 3
    • (c) 7
    • (d) 5
      Answer: (a) 4

69. The foci of an ellipse are located along the y-axis if its equation is of the form:

    • (a) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$
    • (b) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$
    • (c) $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $a > b$
    • (d) $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $b > a$
      Answer: (d) $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $b > a$

70. The length of the latus rectum of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ is:

    • (a) 5
    • (b) 4
    • (c) $6.4$
    • (d) $8$
      Answer: (c) $6.4$

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Hyperbolas

71. The equation of a hyperbola with foci on the x-axis is:

• (a) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
• (b) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
• (c) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
• (d) $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$
  Answer: (b) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

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