1. Length of the chord of the parabola
\[y^{2}=4ax\] passing throught the vertex and making an angle
\[\theta\left(0<\theta<\pi\right)\] with the axis of the parabola is

a) \[4a\mid\cot\theta\mid cosec\theta\]

b) \[2a\mid\cot\theta\mid cosec\theta\]

c) \[a\mid\cot\theta\mid cosec\theta\]

d) \[a \cot^{3} q\]

Explanation:

2. If \[r_{1} \] and \[r_{2} \] are length of two perpendicular chords of the parabola drawn thought the vertex , then value of \[16a^{2} \left[\left(r_{1}r_2^2\right)^{-\frac{2}{3}}+\left(r_1^2r_{2}\right)^{-\frac{2}{3}}\right]\]

is

a) \[\frac{1}{4}\]

b) \[\frac{1}{2}\]

c) 1

d) 2

Explanation:

3. A point P moves such that the difference
between its distance from the origin and from the axis of x is always a constant c.
the locus of P is a

a) straight line having equal intercepts c on the axes

b) circle having its centre at (0, –c/2) and passing
through \[\left(c\sqrt{2},-c/2\right)\]

c) parabola with its vertex at (0, –c/2) and passing
through \[\left(c\sqrt{2},c/2\right)\]

d) none of these .

Explanation: Let the coordinates of P be (h, k)

4. Shortest distance of the point (0, c) from
the parabola \[y=x^{2}\] where \[0\leq c\leq 5\]

is

a) c if \[0\leq c\leq 1/2\]

b) c if \[3\leq c\leq 5\]

c) \[\sqrt{c-1/4}\] if \[1/2\leq c\leq5\] , c if \[0\leq c \leq1/2\]

d) \[\sqrt{c}\]

Explanation: If S is the distance of the point (x, y) on the

5. \[L_{1}\] and \[L_{2}\] are the length of the segments
of any focal chord of the parabola \[y^{2}=x\] , then \[\frac{1}{L_{1}}+\frac{1}{L_{2}}\] is
equal to

a) 2

b) 3

c) 4

d) none of these

Explanation: Any point on the parabola is P(at

^{2}, 2at)

6. The length of the intercept on the normal at the point
\[\left(at^{2},2at\right)\] of the parabola \[y^{2}=4ax\] made by the
circle which is described on the focal distance of the given
point as diameter is

a) \[a\left(1+t^{2}\right)\]

b) \[a\sqrt{1+t^{2}}\]

c) \[\sqrt{a\left(1+t^{2}\right)}\]

d) none of these

Explanation:

7. A line bisecting the ordinate PN of a point
P\[\left(at^{2},2at\right)\] , t > 0, on the parabola \[y^{2}=4ax\] is drawn parallel
to the axis to meet the curve at Q. If NQ meets the tangent
at the vertex at the point T, then the coordinates of T are

a) (0, (4/3)at)

b) (0, 2at)

c) \[\left(\left(1/4\right)at^{2},at\right)\]

d) (0, at)

Explanation: Equation of the line parallel to the axis and

8. If P, Q, R are three points on a parabola
\[y^{2}=4ax\] whose ordinates are in geometrical progression,
then the tangents at P and R meet on

a) the line through Q parallel to x-axis

b) the line through Q parallel to y-axis

c) the line joining Q to the vertex

d) the line joining Q to the focus

Explanation: Let the coordinates of P, Q, R be (at

_{i}

^{2}, 2at

_{i})

9. The tangents at three points A,B,C on the parabola \[y^{2}=4x\] , taken in pairs intersect at the point
P, Q and R. If \[\triangle ,\triangle ' \] be the areas of the triangles ABC and PQR
respectively, then

a) \[\triangle=2\triangle'\]

b) \[\triangle'=2\triangle\]

c) \[\triangle=\triangle'\]

d) none of these

Explanation: Let the coordinate of A, B, C be (t

_{i}

^{2}, 2t

_{i})

10. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola
\[y^{2}=4ax\] is another parabola with directrix

a) x = – a

b) x = – a/2

c) x = 0

d) x = a/2

Explanation: The focus of the parabola y

^{2}= 4ax is S(a, 0), let