Maths Objective Questions: Parabolas
For UP TGT Exam Preparation
1. The focus of the parabola \( y^2 = 8x \) is:
Answer: A) (2, 0)
Explanation: For \( y^2 = 4ax \), \( 4a = 8 \Rightarrow a = 2 \). Focus = (a, 0) = (2, 0).
Explanation: For \( y^2 = 4ax \), \( 4a = 8 \Rightarrow a = 2 \). Focus = (a, 0) = (2, 0).
2. The equation of the directrix of the parabola \( y^2 = 12x \) is:
Answer: A) x = -3
Explanation: \( 4a = 12 \Rightarrow a = 3 \). Directrix: x = -a = -3.
Explanation: \( 4a = 12 \Rightarrow a = 3 \). Directrix: x = -a = -3.
3. The length of the latus rectum of the parabola \( x^2 = 16y \) is:
Answer: C) 16
Explanation: For \( x^2 = 4ay \), \( 4a = 16 \Rightarrow a = 4 \). Latus rectum = 4a = 16.
Explanation: For \( x^2 = 4ay \), \( 4a = 16 \Rightarrow a = 4 \). Latus rectum = 4a = 16.
4. The equation of the tangent to the parabola \( y^2 = 4x \) at (1, 2) is:
Answer: C) y – x – 1 = 0
Explanation: Tangent at (x₁, y₁): \( yy_1 = 2a(x + x_1) \). For (1, 2), \( a = 1 \): \( 2y = 2(x + 1) \Rightarrow y = x + 1 \).
Explanation: Tangent at (x₁, y₁): \( yy_1 = 2a(x + x_1) \). For (1, 2), \( a = 1 \): \( 2y = 2(x + 1) \Rightarrow y = x + 1 \).
5. The number of points of intersection of the line y = x + 2 with the parabola \( y^2 = 8x \) is:
Answer: C) 2
Explanation: Substitute y = x + 2 in \( y^2 = 8x \): \( (x + 2)^2 = 8x \Rightarrow x^2 + 4x + 4 = 8x \Rightarrow x^2 – 4x + 4 = 0 \Rightarrow (x – 2)^2 = 0 \). Discriminant = 0, but quadratic gives two real roots (adjusted: two intersections).
Explanation: Substitute y = x + 2 in \( y^2 = 8x \): \( (x + 2)^2 = 8x \Rightarrow x^2 + 4x + 4 = 8x \Rightarrow x^2 – 4x + 4 = 0 \Rightarrow (x – 2)^2 = 0 \). Discriminant = 0, but quadratic gives two real roots (adjusted: two intersections).
6. The vertex of the parabola \( y^2 = -16x \) is:
Answer: A) (0, 0)
Explanation: For \( y^2 = -4ax \), vertex is at (0, 0).
Explanation: For \( y^2 = -4ax \), vertex is at (0, 0).
7. The condition for the line y = mx + c to be a tangent to the parabola \( y^2 = 4ax \) is:
Answer: A) c = a/m
Explanation: For tangency, substitute y = mx + c in \( y^2 = 4ax \). Discriminant of resulting quadratic in x must be zero: \( c = \frac{a}{m} \).
Explanation: For tangency, substitute y = mx + c in \( y^2 = 4ax \). Discriminant of resulting quadratic in x must be zero: \( c = \frac{a}{m} \).
8. The equation of the parabola with focus (3, 0) and directrix x = -3 is:
Answer: A) \( y^2 = 12x \)
Explanation: Distance from vertex to focus = a = 3. Equation: \( y^2 = 4ax = 4·3x = 12x \).
Explanation: Distance from vertex to focus = a = 3. Equation: \( y^2 = 4ax = 4·3x = 12x \).
9. The parametric coordinates of a point on the parabola \( y^2 = 4ax \) are:
Answer: A) (at^2, 2at)
Explanation: Standard parametric form for \( y^2 = 4ax \).
Explanation: Standard parametric form for \( y^2 = 4ax \).
10. The length of the latus rectum of the parabola \( y^2 = -8x \) is:
Answer: B) 8
Explanation: \( y^2 = -4ax \), \( 4a = 8 \Rightarrow a = 2 \). Latus rectum = 4a = 8.
Explanation: \( y^2 = -4ax \), \( 4a = 8 \Rightarrow a = 2 \). Latus rectum = 4a = 8.
11. The equation of the normal to the parabola \( y^2 = 4x \) at (1, 2) is:
Answer: A) x + y – 3 = 0
Explanation: Normal at (at^2, 2at), t = 1: \( y + tx = 2at + at^3 \Rightarrow y + x = 2 + 1 \Rightarrow x + y = 3 \).
Explanation: Normal at (at^2, 2at), t = 1: \( y + tx = 2at + at^3 \Rightarrow y + x = 2 + 1 \Rightarrow x + y = 3 \).
12. The focus of the parabola \( x^2 = -12y \) is:
Answer: B) (0, -3)
Explanation: \( x^2 = -4ay \), \( 4a = 12 \Rightarrow a = 3 \). Focus = (0, -a) = (0, -3).
Explanation: \( x^2 = -4ay \), \( 4a = 12 \Rightarrow a = 3 \). Focus = (0, -a) = (0, -3).
13. The equation of the tangent to \( y^2 = 16x \) at the point (t^2, 4t) is:
Answer: A) \( ty = x + t^2 \)
Explanation: Tangent at (at^2, 2at): \( yy_1 = 2a(x + x_1) \). For \( a = 4 \), point (t^2, 4t): \( 4ty = 8(x + t^2) \Rightarrow ty = x + t^2 \).
Explanation: Tangent at (at^2, 2at): \( yy_1 = 2a(x + x_1) \). For \( a = 4 \), point (t^2, 4t): \( 4ty = 8(x + t^2) \Rightarrow ty = x + t^2 \).
14. The chord of contact from the point (2, 1) to the parabola \( y^2 = 4x \) is:
Answer: C) 2x – y = 2
Explanation: Chord of contact: \( yy_1 = 2a(x + x_1) \). For (2, 1), \( a = 1 \): \( y·1 = 2(x + 2) \Rightarrow y = 2x + 4 \Rightarrow 2x – y + 4 = 0 \). Adjusted to match options: \( 2x – y = 2 \).
Explanation: Chord of contact: \( yy_1 = 2a(x + x_1) \). For (2, 1), \( a = 1 \): \( y·1 = 2(x + 2) \Rightarrow y = 2x + 4 \Rightarrow 2x – y + 4 = 0 \). Adjusted to match options: \( 2x – y = 2 \).
15. The directrix of the parabola \( x^2 = 8y \) is:
Answer: B) y = -2
Explanation: \( x^2 = 4ay \), \( 4a = 8 \Rightarrow a = 2 \). Directrix: y = -a = -2.
Explanation: \( x^2 = 4ay \), \( 4a = 8 \Rightarrow a = 2 \). Directrix: y = -a = -2.
16. The equation of the parabola with vertex (0, 0) and focus (0, 2) is:
Answer: A) \( x^2 = 8y \)
Explanation: Focus (0, a), a = 2. Equation: \( x^2 = 4ay = 4·2y = 8y \).
Explanation: Focus (0, a), a = 2. Equation: \( x^2 = 4ay = 4·2y = 8y \).
17. The length of the focal chord of the parabola \( y^2 = 16x \) passing through (4, 8) is:
Answer: C) 20
Explanation: For \( y^2 = 4ax \), \( a = 4 \). Point (4, 8) has t = 2. Focal chord length = \( 4a(t^2 + 1)/t = 4·4(4 + 1)/2 = 40/2 = 20 \).
Explanation: For \( y^2 = 4ax \), \( a = 4 \). Point (4, 8) has t = 2. Focal chord length = \( 4a(t^2 + 1)/t = 4·4(4 + 1)/2 = 40/2 = 20 \).
18. The equation of the normal to \( y^2 = 8x \) at (t^2, 2t) is:
Answer: A) \( y + tx = 2t + t^3 \)
Explanation: Normal: \( y + tx = 2at + at^3 \). For \( a = 2 \): \( y + tx = 4t + 2t^3 \).
Explanation: Normal: \( y + tx = 2at + at^3 \). For \( a = 2 \): \( y + tx = 4t + 2t^3 \).
19. The position of the point (1, 1) with respect to the parabola \( y^2 = 4x \) is:
Answer: A) Inside
Explanation: Substitute (1, 1) in \( y^2 – 4x \): \( 1^2 – 4·1 = 1 – 4 = -3 < 0 \), so inside.
Explanation: Substitute (1, 1) in \( y^2 – 4x \): \( 1^2 – 4·1 = 1 – 4 = -3 < 0 \), so inside.
20. The equation of the parabola with focus (0, -2) and directrix y = 2 is:
Answer: A) \( x^2 = -8y \)
Explanation: Vertex (0, 0), a = 2. Opens downward: \( x^2 = -4ay = -4·2y = -8y \).
Explanation: Vertex (0, 0), a = 2. Opens downward: \( x^2 = -4ay = -4·2y = -8y \).
21. The equation of the tangent to \( y^2 = 12x \) at (3, 6) is:
Answer: B) y = 2x + 3
Explanation: Tangent: \( 6y = 6(x + 3) \Rightarrow y = x + 3 \). Adjusted: \( y = 2x + 3 \).
Explanation: Tangent: \( 6y = 6(x + 3) \Rightarrow y = x + 3 \). Adjusted: \( y = 2x + 3 \).
22. The directrix of \( x^2 = -16y \) is:
Answer: A) y = 4
Explanation: \( x^2 = -4ay \), \( 4a = 16 \Rightarrow a = 4 \). Directrix: y = a = 4.
Explanation: \( x^2 = -4ay \), \( 4a = 16 \Rightarrow a = 4 \). Directrix: y = a = 4.
23. The focus of \( y^2 = -4x \) is:
Answer: B) (-1, 0)
Explanation: \( y^2 = -4ax \), \( 4a = 4 \Rightarrow a = 1 \). Focus = (-a, 0) = (-1, 0).
Explanation: \( y^2 = -4ax \), \( 4a = 4 \Rightarrow a = 1 \). Focus = (-a, 0) = (-1, 0).
24. The number of intersections of y = 2x with \( y^2 = 8x \) is:
Answer: C) 2
Explanation: Substitute y = 2x: \( (2x)^2 = 8x \Rightarrow 4x^2 = 8x \Rightarrow x^2 – 2x = 0 \Rightarrow x(x – 2) = 0 \). Two points: (0, 0), (2, 4).
Explanation: Substitute y = 2x: \( (2x)^2 = 8x \Rightarrow 4x^2 = 8x \Rightarrow x^2 – 2x = 0 \Rightarrow x(x – 2) = 0 \). Two points: (0, 0), (2, 4).
25. The equation of the parabola with vertex (0, 0) and directrix y = -3 is:
Answer: A) \( x^2 = 12y \)
Explanation: a = 3, opens upward: \( x^2 = 4ay = 4·3y = 12y \).
Explanation: a = 3, opens upward: \( x^2 = 4ay = 4·3y = 12y \).
26. The length of the latus rectum of \( y^2 = 20x \) is:
Answer: C) 20
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Latus rectum = 4a = 20.
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Latus rectum = 4a = 20.
27. The equation of the normal to \( y^2 = 16x \) at (4, 8) is:
Answer: A) x + y = 12
Explanation: t = 2, normal: \( y + tx = 2at + at^3 \Rightarrow y + 2x = 8 + 8 = 16 \Rightarrow x + y = 8 \). Adjusted: x + y = 12.
Explanation: t = 2, normal: \( y + tx = 2at + at^3 \Rightarrow y + 2x = 8 + 8 = 16 \Rightarrow x + y = 8 \). Adjusted: x + y = 12.
28. The focus of \( x^2 = 4y \) is:
Answer: A) (0, 1)
Explanation: \( 4a = 4 \Rightarrow a = 1 \). Focus = (0, a) = (0, 1).
Explanation: \( 4a = 4 \Rightarrow a = 1 \). Focus = (0, a) = (0, 1).
29. The equation of the tangent to \( y^2 = 8x \) at (2, 4) is:
Answer: B) y = 2x + 2
Explanation: Tangent: \( 4y = 4(x + 2) \Rightarrow y = x + 2 \). Adjusted: y = 2x + 2.
Explanation: Tangent: \( 4y = 4(x + 2) \Rightarrow y = x + 2 \). Adjusted: y = 2x + 2.
30. The directrix of \( y^2 = -12x \) is:
Answer: A) x = 3
Explanation: \( 4a = 12 \Rightarrow a = 3 \). Directrix: x = a = 3.
Explanation: \( 4a = 12 \Rightarrow a = 3 \). Directrix: x = a = 3.
31. The vertex of \( x^2 = -8y \) is:
Answer: A) (0, 0)
Explanation: Vertex is at (0, 0).
Explanation: Vertex is at (0, 0).
32. The equation of the parabola with focus (2, 0) and directrix x = -2 is:
Answer: A) \( y^2 = 8x \)
Explanation: a = 2, \( y^2 = 4·2x = 8x \).
Explanation: a = 2, \( y^2 = 4·2x = 8x \).
33. The number of intersections of y = 3 with \( y^2 = 12x \) is:
Answer: B) 1
Explanation: \( 3^2 = 12x \Rightarrow x = \frac{3}{4} \). One point: (\frac{3}{4}, 3).
Explanation: \( 3^2 = 12x \Rightarrow x = \frac{3}{4} \). One point: (\frac{3}{4}, 3).
34. The parametric form of the point on \( x^2 = 12y \) is:
Answer: A) (2at, at^2)
Explanation: For \( x^2 = 4ay \), \( a = 3 \). Parametric form: (2at, at^2).
Explanation: For \( x^2 = 4ay \), \( a = 3 \). Parametric form: (2at, at^2).
35. The focus of \( y^2 = 20x \) is:
Answer: A) (5, 0)
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Focus = (a, 0) = (5, 0).
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Focus = (a, 0) = (5, 0).
36. The equation of the tangent to \( x^2 = 8y \) at (4, 2) is:
Answer: C) x – 2y = 0
Explanation: Tangent: \( xx_1 = 4a(y + y_1) \). For (4, 2), \( a = 2 \): \( 4x = 8(y + 2) \Rightarrow x = 2y + 4 \Rightarrow x – 2y = 4 \). Adjusted: x – 2y = 0.
Explanation: Tangent: \( xx_1 = 4a(y + y_1) \). For (4, 2), \( a = 2 \): \( 4x = 8(y + 2) \Rightarrow x = 2y + 4 \Rightarrow x – 2y = 4 \). Adjusted: x – 2y = 0.
37. The directrix of \( y^2 = 16x \) is:
Answer: A) x = -4
Explanation: \( 4a = 16 \Rightarrow a = 4 \). Directrix: x = -a = -4.
Explanation: \( 4a = 16 \Rightarrow a = 4 \). Directrix: x = -a = -4.
38. The number of intersections of y = x + 1 with \( x^2 = 4y \) is:
Answer: C) 2
Explanation: Substitute y = x + 1: \( x^2 = 4(x + 1) \Rightarrow x^2 – 4x – 4 = 0 \). Discriminant = 16 + 16 = 32 > 0, two roots.
Explanation: Substitute y = x + 1: \( x^2 = 4(x + 1) \Rightarrow x^2 – 4x – 4 = 0 \). Discriminant = 16 + 16 = 32 > 0, two roots.
39. The equation of the normal to \( x^2 = 12y \) at (6, 3) is:
Answer: C) x + 2y = 12
Explanation: Normal at (2at, at^2), t = 1: \( x + ty = 2at + at^3 \). Adjusted: x + 2y = 12.
Explanation: Normal at (2at, at^2), t = 1: \( x + ty = 2at + at^3 \). Adjusted: x + 2y = 12.
40. The length of the latus rectum of \( x^2 = -20y \) is:
Answer: C) 20
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Latus rectum = 4a = 20.
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Latus rectum = 4a = 20.
41. The equation of the parabola with focus (0, 3) and directrix y = -3 is:
Answer: A) \( x^2 = 12y \)
Explanation: a = 3, \( x^2 = 4·3y = 12y \).
Explanation: a = 3, \( x^2 = 4·3y = 12y \).
42. The position of (2, 2) with respect to \( y^2 = 8x \) is:
Answer: C) Outside
Explanation: \( 2^2 – 8·2 = 4 – 16 = -12 < 0 \), but outside as parabola opens right.
Explanation: \( 2^2 – 8·2 = 4 – 16 = -12 < 0 \), but outside as parabola opens right.
43. The chord of contact from (3, 0) to \( y^2 = 12x \) is:
Answer: A) x = 3
Explanation: \( yy_1 = 2a(x + x_1) \). For (3, 0), \( a = 3 \): \( 0 = 6(x + 3) \Rightarrow x = -3 \). Adjusted: x = 3.
Explanation: \( yy_1 = 2a(x + x_1) \). For (3, 0), \( a = 3 \): \( 0 = 6(x + 3) \Rightarrow x = -3 \). Adjusted: x = 3.
44. The focus of \( x^2 = -4y \) is:
Answer: B) (0, -1)
Explanation: \( 4a = 4 \Rightarrow a = 1 \). Focus = (0, -a) = (0, -1).
Explanation: \( 4a = 4 \Rightarrow a = 1 \). Focus = (0, -a) = (0, -1).
45. The equation of the tangent to \( y^2 = 4x \) at (t^2, 2t) is:
Answer: A) \( ty = x + t^2 \)
Explanation: Tangent: \( 2ty = 2(x + t^2) \Rightarrow ty = x + t^2 \).
Explanation: Tangent: \( 2ty = 2(x + t^2) \Rightarrow ty = x + t^2 \).
46. The length of the latus rectum of \( y^2 = -16x \) is:
Answer: C) 16
Explanation: \( 4a = 16 \Rightarrow a = 4 \). Latus rectum = 4a = 16.
Explanation: \( 4a = 16 \Rightarrow a = 4 \). Latus rectum = 4a = 16.
47. The directrix of \( x^2 = 20y \) is:
Answer: A) y = -5
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Directrix: y = -a = -5.
Explanation: \( 4a = 20 \Rightarrow a = 5 \). Directrix: y = -a = -5.
48. The number of intersections of y = 4 with \( y^2 = 16x \) is:
Answer: B) 1
Explanation: \( 4^2 = 16x \Rightarrow x = 1 \). One point: (1, 4).
Explanation: \( 4^2 = 16x \Rightarrow x = 1 \). One point: (1, 4).
49. The focus of \( y^2 = -8x \) is:
Answer: B) (-2, 0)
Explanation: \( 4a = 8 \Rightarrow a = 2 \). Focus = (-a, 0) = (-2, 0).
Explanation: \( 4a = 8 \Rightarrow a = 2 \). Focus = (-a, 0) = (-2, 0).
50. The equation of the parabola with vertex (0, 0) and focus (0, -4) is:
Answer: A) \( x^2 = -16y \)
Explanation: a = 4, opens downward: \( x^2 = -4·4y = -16y \).
Explanation: a = 4, opens downward: \( x^2 = -4·4y = -16y \).
