Conic sections: parabola ellipse and hyperbola in standard forms - Question Bank

1. What is the condition for the line y = mx + c to be tangent to the hyperbola x^2/a^2 - y^2/b^2 = 1?
A) c^2 = a^2m^2 - b^2
B) c^2 = a^2m^2 + b^2
C) c^2 = b^2m^2 - a^2
D) c^2 = b^2m^2 + a^2
2. What is the condition for the line y = mx + c to be tangent to the ellipse x^2/a^2 + y^2/b^2 = 1?
A) c^2 = a^2m^2 + b^2
B) c^2 = a^2m^2 - b^2
C) c^2 = b^2m^2 + a^2
D) c^2 = b^2m^2 - a^2
3. What is the condition for the line y = mx + c to be tangent to the parabola y^2 = 4ax?
A) c = a/m
B) c = -a/m
C) c = m/a
D) c = -m/a
4. For the hyperbola x^2/a^2 - y^2/b^2 = 1, the transverse axis lies on the:
A) x-axis
B) y-axis
C) Both axes
D) Neither axis
5. For the ellipse x^2/a^2 + y^2/b^2 = 1 (a > b), the major axis lies on the:
A) x-axis
B) y-axis
C) Both axes
D) Neither axis
6. For the parabola y^2 = 4ax, the axis of symmetry is the:
A) x-axis
B) y-axis
C) Line y = x
D) Line y = -x
7. The equation 9x^2 - 4y^2 - 36x + 8y - 4 = 0 represents a hyperbola. What is its center?
A) (2, 1)
B) (-2, -1)
C) (1, 2)
D) (-1, -2)
8. The equation 4x^2 + 9y^2 - 8x + 36y - 4 = 0 represents an ellipse. What is its center?
A) (1, -2)
B) (-1, 2)
C) (2, -1)
D) (-2, 1)
9. The equation x^2 - 4x + 2y + 6 = 0 represents a parabola. What is its vertex?
A) (2, -1)
B) (-2, 1)
C) (1, -2)
D) (-1, 2)
10. What is the length of the latus rectum for the hyperbola x^2/a^2 - y^2/b^2 = 1?
A) 2b^2/a
B) 2a^2/b
C) b^2/a
D) a^2/b
11. What is the length of the latus rectum for the ellipse x^2/a^2 + y^2/b^2 = 1 (a > b)?
A) 2b^2/a
B) 2a^2/b
C) b^2/a
D) a^2/b
12. The standard equation of a hyperbola centered at (h, k) with transverse axis parallel to the x-axis is:
A) (x-h)^2/a^2 - (y-k)^2/b^2 = 1
B) (y-k)^2/a^2 - (x-h)^2/b^2 = 1
C) (x-h)^2/b^2 - (y-k)^2/a^2 = 1
D) (y-k)^2/b^2 - (x-h)^2/a^2 = 1
13. The standard equation of an ellipse centered at (h, k) with major axis parallel to the x-axis is:
A) (x-h)^2/a^2 + (y-k)^2/b^2 = 1 (a > b)
B) (x-h)^2/b^2 + (y-k)^2/a^2 = 1 (a > b)
C) (x-h)^2/a^2 + (y-k)^2/b^2 = 1 (b > a)
D) (x-h)^2/b^2 + (y-k)^2/a^2 = 1 (b > a)
14. The standard equation of a parabola with vertex at (h, k) and opening to the left is:
A) (y-k)^2 = -4a(x-h)
B) (y-k)^2 = 4a(x-h)
C) (x-h)^2 = -4a(y-k)
D) (x-h)^2 = 4a(y-k)
15. Which of the following equations represents a hyperbola?
A) x^2 = 4ay
B) x^2/a^2 + y^2/b^2 = 1
C) y^2 = -4ax
D) x^2/a^2 - y^2/b^2 = 1
16. Which of the following equations represents an ellipse?
A) y^2 = 4ax
B) x^2/a^2 - y^2/b^2 = 1
C) x^2 + y^2 = r^2
D) x^2/a^2 + y^2/b^2 = 1
17. Which of the following equations represents a parabola?
A) x^2 + y^2 = r^2
B) x^2/a^2 + y^2/b^2 = 1
C) y^2 = 4ax
D) x^2/a^2 - y^2/b^2 = 1
18. The vertices of the hyperbola y^2/a^2 - x^2/b^2 = 1 are at:
A) (0, ±a)
B) (±a, 0)
C) (0, ±b)
D) (±b, 0)
19. The equation of the directrix for the parabola x^2 = 4ay is:
A) y = -a
B) y = a
C) x = -a
D) x = a
20. What is the latus rectum of the parabola x^2 = 4ay?
A) 4a
B) 2a
C) a
D) a/2
21. For the hyperbola x^2/a^2 - y^2/b^2 = 1, the foci are located at:
A) (±c, 0) where c^2 = a^2 + b^2
B) (0, ±c) where c^2 = a^2 + b^2
C) (±c, 0) where c^2 = b^2 - a^2
D) (0, ±c) where c^2 = b^2 - a^2
22. The equation y^2/a^2 - x^2/b^2 = 1 represents a hyperbola with the transverse axis along the:
A) y-axis
B) x-axis
C) Both axes
D) Neither axis
23. For the ellipse x^2/a^2 + y^2/b^2 = 1 (a > b), the foci are located at:
A) (±c, 0) where c^2 = a^2 - b^2
B) (0, ±c) where c^2 = a^2 - b^2
C) (±c, 0) where c^2 = b^2 - a^2
D) (0, ±c) where c^2 = b^2 - a^2
24. The standard equation of an ellipse centered at the origin with the major axis along the y-axis is:
A) x^2/b^2 + y^2/a^2 = 1 (a > b)
B) x^2/a^2 + y^2/b^2 = 1 (a > b)
C) x^2/a^2 + y^2/b^2 = 1 (b > a)
D) x^2/b^2 + y^2/a^2 = 1 (b > a)
25. The directrix of the parabola y^2 = 4ax is the line:
A) x = -a
B) x = a
C) y = -a
D) y = a
26. A conic section where the eccentricity e > 1 is called a:
A) Hyperbola
B) Parabola
C) Ellipse
D) Circle
27. A conic section where the eccentricity 0 < e < 1 is called a:
A) Ellipse
B) Parabola
C) Hyperbola
D) Circle
28. A conic section where the eccentricity e = 1 is called a:
A) Parabola
B) Ellipse
C) Hyperbola
D) Circle
29. What are the equations of the asymptotes for the hyperbola x^2/a^2 - y^2/b^2 = 1?
A) y = ±(b/a)x
B) y = ±(a/b)x
C) y = ±bx
D) y = ±ax
30. What are the coordinates of the vertices of the hyperbola x^2/a^2 - y^2/b^2 = 1?
A) (±a, 0)
B) (0, ±a)
C) (±b, 0)
D) (0, ±b)
31. For a hyperbola, the eccentricity 'e' always satisfies which condition?
A) e > 1
B) 0 < e < 1
C) e = 1
D) e = 0
32. What is the eccentricity of a hyperbola?
A) e = c/a
B) e = a/c
C) e = c/b
D) e = b/c
33. What is the relationship between a, b, and c (distance from center to focus) in a hyperbola?
A) c^2 = a^2 + b^2
B) c^2 = a^2 - b^2
C) c^2 = b^2 - a^2
D) c = a + b
34. In the hyperbola x^2/a^2 - y^2/b^2 = 1, what is the length of the transverse axis?
A) 2a
B) 2b
C) a
D) b
35. What is the standard equation of a hyperbola centered at the origin with the transverse axis along the x-axis?
A) x^2/a^2 - y^2/b^2 = 1
B) y^2/a^2 - x^2/b^2 = 1
C) x^2/b^2 - y^2/a^2 = 1
D) y^2/b^2 - x^2/a^2 = 1
36. What are the coordinates of the foci of the ellipse x^2/a^2 + y^2/b^2 = 1 (a > b)?
A) (±c, 0)
B) (0, ±c)
C) (±a, 0)
D) (0, ±b)
37. For an ellipse, the eccentricity 'e' always satisfies which condition?
A) 0 < e < 1
B) e = 1
C) e > 1
D) e = 0
38. What is the eccentricity of an ellipse?
A) e = c/a
B) e = a/c
C) e = c/b
D) e = b/c
39. What is the relationship between a, b, and c (distance from center to focus) in an ellipse where a > b?
A) c^2 = a^2 - b^2
B) c^2 = b^2 - a^2
C) c^2 = a^2 + b^2
D) c = a + b
40. For the same ellipse, what is the length of the minor axis?
A) 2b
B) 2a
C) b
D) a
41. In the ellipse x^2/a^2 + y^2/b^2 = 1 (a > b), what is the length of the major axis?
A) 2a
B) 2b
C) a
D) b
42. What is the standard equation of an ellipse centered at the origin with the major axis along the x-axis?
A) x^2/a^2 + y^2/b^2 = 1 (a > b)
B) x^2/b^2 + y^2/a^2 = 1 (a > b)
C) x^2/a^2 + y^2/b^2 = 1 (b > a)
D) x^2/b^2 + y^2/a^2 = 1 (b > a)
43. For the parabola x^2 = 4ay, what are the coordinates of the focus?
A) (0, a)
B) (a, 0)
C) (0, -a)
D) (-a, 0)
44. The equation x^2 = -12y represents a parabola opening in which direction?
A) Upwards
B) Downwards
C) Left
D) Right
45. What is the length of the latus rectum of a parabola?
A) 2a
B) 4a
C) a
D) a/2
46. If the vertex of a parabola is at (h, k) and it opens upwards, what is its standard equation?
A) (x-h)^2 = 4a(y-k)
B) (y-k)^2 = 4a(x-h)
C) (x-h)^2 = -4a(y-k)
D) (y-k)^2 = -4a(x-h)
47. What is the equation of the directrix for the parabola y^2 = 4ax?
A) x = a
B) x = -a
C) y = a
D) y = -a
48. For a parabola y^2 = 4ax, what are the coordinates of the focus?
A) (a, 0)
B) (0, a)
C) (-a, 0)
D) (0, -a)
49. What is the standard equation of a parabola with its vertex at the origin and opening to the right?
A) y^2 = 4ax
B) x^2 = 4ay
C) y^2 = -4ax
D) x^2 = -4ay
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