Question

$TP$ and $TQ$ are tangents to the parabola $y^2=4ax$ at $P$ and $Q.$ If the chord $PQ$ passes through the fixed point $(-a,b),$ then the locus of $T$ is

• A. $ay=2b(x-b)$
• B. $bx=2a(y-a)$
• C. $by=2a(x-a)$
• D. $ax=2b(y-b)$

Answer 1

The correct option is C $by=2a(x-a)$

Let $T\equiv (h,k).$
Chord of contact of $(h,k)$ is given by
$ky=\frac{4a(x+h)}{2}$
$\Rightarrow ky=2a(x+h)$

As it passes through $(-a,b),$
$bk=2a(-a+h)$
Hence, required locus is $by=2a(x-a)$