JEE Main - Conic sections: parabola ellipse and hyperbola in standard forms Online Test (MCQ Quiz)

1. 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

2. 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)

3. 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

4. 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)

5. What is the length of the latus rectum of a parabola?

A) 2a B) 4a C) a D) a/2

6. The equation x^2 = -12y represents a parabola opening in which direction?

A) Upwards B) Downwards C) Left D) Right

7. 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)

8. 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)

9. 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

10. For the same ellipse, what is the length of the minor axis?

A) 2b B) 2a C) b D) a

11. 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

12. What is the eccentricity of an ellipse?

A) e = c/a B) e = a/c C) e = c/b D) e = b/c

13. For an ellipse, the eccentricity 'e' always satisfies which condition?

A) 0 < e < 1 B) e = 1 C) e > 1 D) e = 0

14. 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)

15. 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

16. 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

17. 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

18. What is the eccentricity of a hyperbola?

A) e = c/a B) e = a/c C) e = c/b D) e = b/c

19. For a hyperbola, the eccentricity 'e' always satisfies which condition?

A) e > 1 B) 0 < e < 1 C) e = 1 D) e = 0

20. 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)

21. 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

22. A conic section where the eccentricity e = 1 is called a:

A) Parabola B) Ellipse C) Hyperbola D) Circle

23. A conic section where the eccentricity 0 < e < 1 is called a:

A) Ellipse B) Parabola C) Hyperbola D) Circle

24. A conic section where the eccentricity e > 1 is called a:

A) Hyperbola B) Parabola C) Ellipse D) Circle

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. 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)

27. 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

28. 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

29. 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

30. What is the latus rectum of the parabola x^2 = 4ay?

A) 4a B) 2a C) a D) a/2

31. The equation of the directrix for the parabola x^2 = 4ay is:

A) y = -a B) y = a C) x = -a D) x = a

32. 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)

33. 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

34. 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

35. 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

36. 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)

37. 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)

38. 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

39. 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

40. 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

41. 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)

42. 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)

43. 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)

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

Conic sections: parabola ellipse and hyperbola in standard forms - Online Test

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