Question

$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Tangents are drawn from the point}(2,3)& \text{(P)}& (9,-6)\\ & \text{to the parabola}{y}^2=4x.\text{Then point(s) of}& & \\ & \text{contact is (are)}& & \\ \text{(B)}& \text{From a point}P\text{on the circle}{x}^2+y^2=5,& \text{(Q)}& (1,2)\\ & \text{the equation of chord of contact to the}& & \\ & \text{parabola}{y}^2=4x\text{is}y=2(x-2).\text{Then}& & \\ & \text{the coordinates of}P\text{are}& & \\ \text{(C)}& P(4,-4),Q\text{are points on parabola}& \text{(R)}& (-2,1)\\ & y^2=4x\text{such that area of}△POQ\text{is}6& & \\ & \text{sq. units where}O\text{is the vertex. Then}& & \\ & \text{coordinates of}Q\text{may be}& & \\ \text{(D)}& \text{The common chord of circle}{x}^2+y^2=5& \text{(S)}& (4,4)\\ & \text{and parabola}6y=5x^2+7x\text{will pass}& & \\ & \text{through point(s)}& & \\ & & \text{(T)}& (-2,2)\end{array}$

Which of the following is the only CORRECT combination?

• A. $\text{(A)}\to \text{(P)},\text{(S)}$
• B. $\text{(A)}\to \text{(Q)},\text{(T)}$
• C. $\text{(B)}\to \text{(R)}$
• D. $\text{(B)}\to \text{(Q)},\text{(R)}$

Answer 1

The correct option is C $\text{(B)}\to \text{(R)}$

$\left(A\right)$
Let the point on the parabola be $(t^2,2t)$
The equation of the tangent is,
$2ty=2(x+t^2)$
$\Rightarrow ty=x+t^2$
It passes through the point $(2,3),$ then
$3t=2+t^2\Rightarrow t=1,2$
The point of contact can be $(1,2)$ or $(4,4)$
$\left(A\right)\to \left(Q\right),\left(S\right)$

$\left(B\right)$
Let a point on the circle be $P(x_1,y_1).$
Then the chord of contact of the parabola w.r.t. $P$ is
$y{y}_1=2(x+x_1)$
Comparing with $y=2(x-2)$, we have
$y_1=1$ and $x_1=-2$,
which also satisfy the circle.
$\left(B\right)\to \left(R\right)$