Thank you for reporting, we will resolve it shortly
Q.
If the chord of contact of tangents from a point $P$ to the parabola $y^{2}=4 a x$ touches the parabola $x^{2}=4 b y$, then the locus of $P$ is
Conic Sections
Solution:
The chord of contact of the parabola $y^{2}=4 a x$ w.r.t. point $P\left(x_{1}, y_{1}\right)$ is
$yy_1 = 2a(x + x_1)$ .... (1)
This line touches the parabola $x^{2}=4 b y$.
Solving (1) with parabola, we have
$x^{2}=4 b\left[\frac{2 a}{y_{1}}\left(x+x_{1}\right)\right]$
or $ y_{1} x^{2}-8 a b x-8 a b x_{1}=0$
According to the question, this equation must have equal roots.
Therefore,
$D=0$
or $64 a^{2} b^{2}+32 a b x_{1} y_{1}=0$
or $ x_{1} y_{1}=-2 a b$ or $x y=-2 a b$
which is the required locus.