$\text{ If the chord of contact of tangents from a point }P\text{ to the parabola }{y}^2=4ax\text{ touches the parabola }{x}^2=4by,\text{ then the }$$\text{ locus of }P\text{ is }$

$\text{ The chord of contact of the parabola }{y}^2=4ax\text{ w.r.t. point }P\left(x_1,y_1\right)\text{ is }$

$y{y}_1=2a\left(x+x_1\right)-------\left(1\right)$

$\text{ This line touches the parabola }{x}^2=4by\text{ . }$

$\text{ Solving (1) with parabola, we have }$

$x^2=4b\left[\frac{2a}{y_1}\left(x+x_1\right)\right]$

$\text{ or }{y}_1{x}^2-8abx-8ab{x}_1=0$

According to the question, this equation must have equal roots.
Therefore, D = 0

$\begin{array}{l}\text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}64a^2{b}^2+32ab{x}_1{y}_1=0\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{x}_1{y}_1=-2ab\text{ or }xy=-2ab\end{array}$

which is the required locus.