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  4. A tangent and a normal are drawn at the point P (2,-4) on the parabola y 2=8 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:

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Q. A tangent and a normal are drawn at the point $P (2,-4)$ on the parabola $y ^{2}=8 x$, 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:

JEE MainJEE Main 2021Conic Sections

Solution:

image
Equation of tangent at $(2,-4)( T =0)$
$-4 y=4(x+2)$
$x+y+2=0 \,\,\, ...(1)$
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 $A(-2,8)$
Given $AQBP$ is a sq.
image
$\Rightarrow m _{ AQ } .m _{ AP }=-1$
$\Rightarrow \left(\frac{b}{a+2}\right)\left(\frac{4}{-4}\right)=-1 $
$\Rightarrow a+2=b \,\,\, ...(1)$
Also $PQ$ must be parallel to x-axis thus
$\Rightarrow b =-4$
$\therefore a =-6$
Thus $2 a+b=-16$