The locus of the point of intersection of tangents to the hyperbola $4x^2-9y^2=36$ which meet at a constant angle $\pi /4,$ is:

Let the point of intersection of tangents be $P\left(x_1,y_1\right)$,

Then the equation of pair of tangents from $P\left(x_1,y_1\right)$ to the 

given hyperbola is $\left(4x^2-9y^2-36\right)\left(4x_1^2-9y_1^2-36\right)$$={\left[4x_{1}x-9y_{1}y-36\right]}^2$                                         ….(i)

From $S{S}_1=T^2$ 

or $x^2\left(y_1^2+4\right)+2x_1{y}_{1}xy+y^2\left(x_1^2-9\right)+\dots =0 \dots \left(ii\right)$

Since angle between the tangents $\pi /4,$

$\therefore$ $\tan (\pi /4)=\frac{2\sqrt{\left[x_1^2{y}_1^2-\left(y_1^2+4\right)\left(x_1^2-9\right)\right]}}{y_1^2+4+x_1^2-9}$

Hence locus of $P\left(x_1,y_1\right)$ is ${\left(x^2+y^2-5\right)}^2$$=4\left(9y^2-4x^2+36\right)$.