Practice Test – 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
Click to See Answer :
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
Click to See Answer :
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)
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
8. The graph represented by equations x = sin2t , y = 2 cost is
(A) hyperbola
(B) sine graph
(C) parabola
(D) straight line
Click to See Answer :
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
Click to See Answer :
10. If the normal 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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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)
Click to See Answer :
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) $\large (y+\frac{8}{9})^2 = \frac{4}{9}(x + \frac{2}{9})$
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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}
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
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
Click to See Answer :
Click to See All Answers :
11. C 12. B 13. A 14. C 15. A 16. B 17. C 18. D 19. A 20. B
21. B 22. A 23. A 24. D 25. A 26. B 27. A 28. A 29. C 30. C