$$\begin{gathered} \mathrm{If} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{asymptotes} \mathrm{of} \mathrm{the} \mathrm{hyperbola} {\mathrm{x}}^2/{\mathrm{a}}^2-{\mathrm{y}}^2/{\mathrm{b}}^2=1 \mathrm{is} {120}^0\mathrm{and} \mathrm{product} \mathrm{of} \mathrm{perpendiculars} \\ \mathrm{drawn} \mathrm{from} \mathrm{foci} to \mathrm{any} \mathrm{tangent} \mathrm{is} 9, \mathrm{then} \mathrm{the} \mathrm{locus} \mathrm{of} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} \mathrm{perpendicular} \mathrm{tangents} \mathrm{of} \mathrm{the} \\ \mathrm{hyperbola} \mathrm{can} \mathrm{be} \end{gathered}$$

$$\begin{gathered} Given hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ Given that p\mathrm{roduct} \mathrm{of} \mathrm{perpendicular} \mathrm{from} \mathrm{foci}={\mathrm{b}}^2=9 \\ and also given a\mathrm{ngle} \mathrm{between} \mathrm{asymptotes} =120 or {60}^0 \\ \Rightarrow 2{\sec }^{-1}\left(\mathrm{e}\right)={60}^0 \\ \Rightarrow e=\frac{2}{\sqrt{3}} \\ Now b^2=a^2(e^2-1) \\ \Rightarrow {\mathrm{a}}^2=27 \\ Now locus of point of intersection of perpendicular \tan gents is director circle \\ \mathrm{Equation} \mathrm{of} \mathrm{directrix} \mathrm{circle} is x^2+{\mathrm{y}}^2=18 \\ At \theta ={120}^{0 },director circle does not exist. \end{gathered}$$