# A tangent and a normal are drawn

Question:

A tangent and a normal are drawn at the point $\mathrm{P}(2,-4)$ on the parabola $\mathrm{y}^{2}=8 \mathrm{x}$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If $Q(a, b)$ is a point such that $A Q B P$ is a square, then $2 a+b$ is equal to :

1. -16

2. -18

3. -12

4. -20

Correct Option: 1

Solution:

Equation of tangent at $(2,-4)(\mathrm{T}=0)$

$-4 y=4(x+2)$

$x+y+2=0$..(1)

equation of normal

$x-y+\lambda=0$

$\downarrow(2,-4)$

$\lambda=-6$

thus $x-y=6 \ldots(2)$ equation of normal

POI of (1) \& $x=-2$ is $A(-2,0)$

POI of $(2) \& x=-2$ is $\mathrm{A}(-2,8)$

Given $\mathrm{AQBP}$ is a sq.

$\Rightarrow \mathrm{m}_{\mathrm{AQ}} \cdot \mathrm{m}_{\mathrm{AP}}=-1$

$\Rightarrow\left(\frac{\mathrm{b}}{\mathrm{a}+2}\right)\left(\frac{4}{-4}\right)=-1 \Rightarrow \mathrm{a}+2=\mathrm{b}$

Also PQ must be parallel to $x$-axis thus

$\Rightarrow \mathrm{b}=-4$

$\therefore \mathrm{a}=-6$

Thus $2 a+b=-16$

27 August 2021, Time 9 AM to 12 PM (Shift 1)