## Parabola Questions and Answers Part-1

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(at2 , 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 (ati2 , 2ati)

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

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