Thank you for reporting, we will resolve it shortly
Q.
Tangents are drawn at the end points of a normal chord of the parabola $y^{2}=4ax$ . The locus of their point of intersection is
NTA AbhyasNTA Abhyas 2022
Solution:
Let, $P\left(a t_{1}^{2} , 2 a t_{1}\right)$ and $Q\left(a t_{2}^{2} , 2 a t_{2}\right)$ be the end points of a normal chord of the parabola $y^{2}=4ax$ such that $PQ$ is normal at $P.$ Then,
$t_{2}=-t_{1}-\frac{2}{t_{1}}$ …(i)
Let, $\left(h , k\right)$ be the point of intersection of tangents to the parabola at $P$ and $Q.$ Then,
$h=at_{1}t_{2}$ and $k=a\left(t_{1} + t_{2}\right)$
$\Rightarrow h=at_{1}\left(- t_{1} - \frac{2}{t_{1}}\right)$ and $k=\frac{- 2 a}{t_{1}}.$ [Using (i)]
$\Rightarrow h=-a\left(t_{1}^{2} + 2\right)$ and $t_{1}=\frac{- 2 a}{k}$
$\Rightarrow h=-a\left(\frac{4 a^{2}}{k^{2}} + 2\right)$
$\Rightarrow \left(h + 2 a\right)k^{2}=-4a^{3}$
Hence, the locus of $\left(h , k\right)$ is
$\left(x + 2 a\right)y^{2}=-4a^{3}$
or $\left(x + 2 a\right)y^{2}+4a^{3}=0$