MCQ : Parabola

LEVEL – I
1. The equation of the tangent at the vertex of the parabola x2 + 4x + 2y = 0 is

(A) x = –2

(B) x = 2

(C) y = 2

(D) y = –2.

2. BC is latus rectum of a parabola y2 = 4ax and A is its vertex, then minimum length of projection of BC on a tangent drawn in portion BAC is

(A) a

(B) 2√a

(C) 2a

(D) 3a

3. The coordinates of the point on the parabola y = x2 + 7x + 2 , which is nearest to the straight line y = 3x – 3 are

(A) (-2, -8)

(B) (1, 10)

(C) (2, 20)

(D) (-1, -4)

4. The angle between tangents drawn form the point (3 , 4) to the parabola y2 – 2y + 4x = 0 is

(A) tan-1(8√5/7)

(B) tan-1(12/√5)

(C) tan-1(√5/7)

(D) none of these

5. If the line x + y – 1 = 0 touches the parabola y2 = kx , then the value of k is

(A) 4

(B) –4

(C) 2

(D) –2

6. If (3t12-6t1) represents the feet of the normals to the parabola y2 = 12x from (1, 2), then Σ1/t1 is

(A) – 5/2

(B) 3/2

(C) 6

(D) –3

7. Two parabolas y2 = 4a(x – λ1) and x2 = 4a(y – λ2) always touch each other (λ1, λ2 being variable parameters). Then their point of contact lies on a

(A) straight line

(B) circle

(C) parabola

(D) hyperbola

8. The graph represented by equations x = sin2t , y = 2 cost is

(A) hyperbola

(B) sine graph

(C) parabola

(D) straight line

9. If 2 and 3 are the length of the segments of any focal chord of a parabola y2 = 4ax, then value of 2a is

(A) 13/5

(B) 12/5

(C) 11/5

(D) none of these

10. If the normals drawn at the end points of a variable chord PQ of the parabola y2 = 4ax intersect at parabola, then the locus of the point of intersection of the tangent drawn at the points P and Q is

(A) x + a = 0

(B) x – 2a = 0

(C) y2 – 4x + 6 = 0

(D) none of these

ANSWER:

1. C  2. B  3. A  4. A  5. B  6. A  7. D  8. C  9. B  10. B 

LEVEL – I
11. If it is not possible to draw any tangent from the point (1/4, 1) to the parabola y2 = 4x cosθ + sin2θ , then θ belongs to

(A) [-π/2 π/2]

(B) [-π/2 π/2] – {0}

(C) (-π/2 π/2) – {0}

(D) none of these

12. The number of focal chord(s) of length 4/7 in the parabola 7y2 = 8x is

(A) 1

(B) zero

(C) infinite

(D) none of these

13. The ends of line segment are P (1, 3) and Q (1, 1). R is a point on the line segment PQ such that PR : RQ = 1 : λ . If R is an interior point of parabola y2 = 4x, then

(A) λ ∈ (0, 1)

(B) λ ∈ (-3/5 , 1)

(C) λ ∈ (1/2 , 3/5)

(D) none of these

14. A set of parallel chords of the parabola y2 = 4ax have their mid points on

(A) any straight line through the vertex

(B) any straight line through the focus

(C) a straight line parallel to the axis

(D) another parabola

15. The equation of the line of the shortest distance between the parabola y2 = 4x and the circle x2 + y2 – 4x – 2y + 4 = 0 is

(A) x + y = 3

(B) x – y = 3

(C) 2x + y = 5

(D) none of these

16. If normals are drawn from the extremities of the latus rectum of a parabola then normals are

(A) parallel to each other

(B) perpendicular to each other

(C) intersect at the 450

(D) none of these

17. The triangle formed by the tangent to the parabola y = x2 at the point whose abscissa is k where k ∈ [1, 2] the y-axis and the straight line y = k2 has greatest area if k is equal to

(A) 1

(B) 3

(C) 2

(D) none of these

18. A parabola y2 = 4ax and x2 = 4by intersect at two points. A circle is passed through one of the intersection point of these parabola and touch the directrix of first parabola then the locus of the centre of the circle is

(A) straight line

(B) ellipse

(C) circle

(D) parabola

19. A circle with centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is

(A) (p/2 , p)

(B) (p/2 , 2p)

(C) (-p/2 , p)

(D) (-p/2 , -p)

20. The locus of a point divides a chord of slope 2 of the parabola y2 = 4x internally in the ratio 1 : 2 is

(A) $\large (y+\frac{8}{9})^2 = \frac{4}{9}(x-\frac{2}{9})$

(B) $\large (y-\frac{8}{9})^2 = \frac{4}{9}(x-\frac{2}{3})$

(C) $\large (y-\frac{8}{9})^2 = \frac{4}{9}(x + \frac{2}{9})$

(D) $\large (y+\frac{8}{9})^2 = \frac{4}{9}(x + \frac{2}{9})$

ANSWER:

11. C  12. B  13. A  14. C  15. A  16. B  17. C  18. D  19. A  20. B 

LEVEL – I

21. The point (1, 2) is one extremity of focal chord of parabola y2 = 4x. The length of this focal chord is

(A) 2

(B) 4

(C) 6

(D) none of these

22. If AFB is a focal chord of the parabola y2 = 4ax and AF = 4, FB = 5, then the latus-rectum of the parabola is equal to

(A) 80/9

(B) 9/80

(C) 9

(D) 80

23. If three normals can be drawn from (h, 2) to the parabola y2 = -4x, then

(A) h < -2

(B) h > 2

(C) –2 < h < 2

(D) h is any real number

24. If the line y – √ x +3 = 0 cuts the parabola y2 = x + 2 at A and B, and if P≡ (√3 , 0), then PA. PB is equal to

(A) 2(√3+2)/3

(B) 4√3/2

(C) 4(2-√3)/3

(D) 4(√3+2)/3

25. If the normal to the parabola y2 = 4ax at the point (at2, 2at) cuts the parabola again at (aT2, 2aT), then

(A) T2 ≥ 8

(B) T ∈ (- ∞, -8) ∪ (8, ∞)

(C) –2 ≤ T ≤ 2

(D) T2 < 8

26. The locus of point of intersection of any tangent to the parabola y2 = 4a (x – 2) with a line perpendicular to it and passing through the focus, is

(A) x = 1

(B) x = 2

(C) x = 0

(D) none of these

27. The set of points on the axis of the parabola y2 – 4x – 2y + 5 = 0 from which all the three normals to the parabola are real is

(A) {(x, 1) : x ≥ 3}

(B) {(x, -1) : x ≥ 1}

(C) {(x, 3) : x ≥ 1}

(D) {(x, -3) : x ≥ 3}

28. If at x = 1, y = 2x is tangent to the parabola y = ax2 + bx + c, then respective values of a, b, c are

(A) 1/2 , 1, 1/2

(B) 1, 1/2 , 1/2

(C) 1/2 , 1/2 , 1

(D) none of these

29. If the segment intercepted by the parabola y2 = 4ax with the line lx + my + n = 0 subtends a right angle at vertex the

(A) al + n = 0

(B) 4am + n = 0

(C) 4al + n = 0

(D) none of these

30. The two parabolas y2 = 4x and x2 = 4y intersect at a point P, whose abscissae is not zero, such that

(A) they both touch each other at P

(B) they cut at right angles at P

(C) the tangents to each curve at P make complementary angles with the x-axis

(D) none of these

ANSWER:

21. B  22. A  23. A  24. D  25. A  26. B  27. A  28. A  29. C  30. C 

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