Embibe Experts Solutions for Chapter: Parabola, Exercise 1: JEE Advanced Paper 2 - 2014

Let $E$ denote the parabola $y^2=8x.$ Let $P=\left(-2,4\right)$, and let $Q$ and $Q^{\text{'}}$ be two distinct points on $E$ such that the lines $PQ$ and $P{Q}^{\text{'}}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE?

Let the circles $C_1:x^2+y^2=9$ and $C_2:{\left(x-3\right)}^2+{\left(y-4\right)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ satisfies the following conditions:
$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$
$\left(ii\right) C_1$ and $C_2$ both lie inside $C_3,$ and
$\left(iii\right) C_3$ touches $C_1$ at $M$ and $C_2$ at $N.$
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W,$ and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2=8\alpha y.$
There are some expression given in the List- $\mathrm{I}$ whose values are given in List- $\mathrm{I}\mathrm{I}$ below:
 

Let the circles $C_1:x^2+y^2=9$ and $C_2:{\left(x-3\right)}^2+{\left(y-4\right)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ satisfies the following conditions:
$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$
$\left(ii\right) C_1$ and $C_2$ both lie inside $C_3,$ and
$\left(iii\right) C_3$ touches $C_1$ at $M$ and $C_2$ at $N.$
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W,$ and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2=8\alpha y.$
There are some expression given in the List- $I$ whose values are given in List- $\mathrm{I}\mathrm{I}$ below:
 

Let $P$ be the point on the parabola $y^2=4x$ which is at the shortest distance from the center $S$ of the circle $x^2+y^2-4x-16y+64=0.$ Let $Q$ be the point on the circle dividing the line segment $SP$ internally. Then -

The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P,$ $Q$ and the parabola at the points $R, S$. Then the area of the quadrilateral $SRPQ$ is

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $\left(h,k\right)$, then which of the following is (are) possible value(s) of $p,h$ and $k$ ?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x\mathrm{ }$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\mathrm{\Delta }OPQ$ is $3\sqrt{2}$ sq. units, then which of the following is (are) the coordinates of $P$?