If the chord of contact of tangents from a point P to the parabola $y^2=4ax$ touches the parabola $x^2=4by$ then

the locus of P is 

If P be (h, k) then  chord of contact is $ky=2a(x+h)$

It touches the parabola $x^2=4by.$

Hence it will cut it in two coincident points. Eliminating y we get 

$x^2=4b\cdot \frac{2a(x+h)}{k}$

$\Rightarrow$ $k{x}^2-8abx-8abh=0$

It must have equal roots

$\therefore 64a^2{b}^2+32abhk=0\Rightarrow hk+2ab=0$

Hence the locus of P is $xy=-2ab$ which is a hyperbola.