1. Equation of a common tangent to the
curves \[y^{2}=8x\] and xy = – 1 is

a) 3y = 9x + 2

b) y = 2x + 1

c) 2y = x + 8

d) y = x + 2

Explanation: Equation of a tangent at (at

^{2}, 2at) to y

^{2}= 8x is ty = x + at

^{2}where 4a = 8 i.e. a = 2

2. The tangent at the point \[P\left(x_{1},y_{1}\right)\] to the parabola \[y^{2}=4ax\] meets the parabola \[y^{2}=4a\] (x + b) at Q
and R, the coordinates of the mid-point of QR are

a) \[\left(x_{1}-a,y_{1}+b\right)\]

b) \[\left(x_{1},y_{1}\right)\]

c) \[\left(x_{1}+b,y_{1}+a\right)\]

d) \[\left(x_{1}-b,y_{1}-b\right)\]

Explanation: Equation of the tangent at P (x

_{1}, y

_{1}) to the

3. Consider a parabola y^{2} = 4ax, the length
of focal chord is l and the length of the perpendicular from
vertex to the chord is p then

a) l . p is constant

b) \[l p^{2}\] is constant

c) \[l^{2}\] p is constant

d) none of these

Explanation: Let P(at

^{2}, 2at) and Q(a/t

^{2}, –2a/t) be a focal

4. Tangent are drawn to a parabola from a point T. If P, Q are the points of contact, then perpendicular
distance from P, T and Q upon the tangent at the vertex of
the parabola are in

a) A.P

b) G.P

c) H.P

d) none of these

Explanation: Let P(at

_{1}

^{2}, 2at), Q(at

_{2}

^{2}, 2at

^{2}), then the point of intersection of the tangents t

_{1}y = x + at

_{1}

^{2}and t

_{2}y = x + at

_{2}

^{2}is T(at

_{1}t

_{2}, a(t

_{1}+ t

_{2}))

5. Chords of the parabola \[y^{2}+4y=\frac{4}{3}x-\frac{16}{3}\] which subtend right angle at the vertex pass through

a) (7/3, –2)

b) (1/3, 0)

c) (4/3, 0)

d) (0, 4/3)

Explanation: Equation of the parabola is

6. The locus of the vertices of the family of parabolas \[y=\frac{a^{3}x^{2}}{3}+\frac{a^{2}x}{2}-2a\]

is

a) \[xy=\frac{105}{64}\]

b) \[xy=\frac{3}{4}\]

c) \[xy=\frac{35}{16}\]

d) \[xy=\frac{64}{105}\]

Explanation: Equation of the parabola is

7. Consider the two curves
\[c_{1}:y^{2}=4x;c_{2}:x^{2} +y^{2} -6x+1\]

then

a) \[c_{1}\] and \[c_{2}\] touch each other only at one point

b) \[c_{1}\] and \[c_{2}\] touch each other exactly at two points

c) \[c_{1}\] and \[c_{2}\] intersect (but do not touch) at exactly two point

d) \[c_{1}\] and \[c_{2}\] neither intersect nor touch each other

Explanation: Solving the two equations, we get

8. If AB is a focal chord of the parabola
\[y^{2}=4ax\] with focus F, then harmonic mean FA and FB is

a) a

b) 2a

c) 4a

d) 8a

Explanation: End points of the focal chord be

9. Length of the shortest normal chord of
the parabola \[y^{2}=4x\] is

a) 6

b) \[6\sqrt{3}\]

c) 1

d) \[3\sqrt{3}\]

Explanation: Equation of normal to the parabola y

^{2}= 4x at

10. Let \[\left(x_{1},y_{1}\right)\] be a point outside the parabola
\[y^{2}=4ax\] . Length of the chord of contact of tangents drawn
from point \[\left(x_{1},y_{1}\right)\] to \[y^{2}=4ax\] is

a) \[\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(y_1^2+4a^{2}\right)}\]

b) \[\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(x_1^2+a^{2}\right)}\]

c) \[\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(x_1^2+4a^{2}\right)}\]

d) \[\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(y_1^2+a^{2}\right)}\]

Explanation: Let the tangets at the point A (at

_{1}

^{2}, 2at

_{1}) and B(at

_{2}

^{2},2at

_{2}) pass through (x

_{1}, y

_{1}) ,