The point of intersection of the perpendicular tangents drawn to the ellipse $4{\mathrm{x}}^2+9{\mathrm{y}}^2 = 36$ lies on the circle

Locus of point of intersection of perpendicular tangents is director circle 

given equation $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$

equation  of director circle is$$\begin{gathered} {\mathrm{x}}^2+y^2=a^2+b^2 \\ \Rightarrow {\mathrm{x}}^2+y^2=9+4=13 \end{gathered}$$