Tangents to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ make angles ${\mathrm{\theta }}_1$ and ${\mathrm{\theta }}_2$ with the major axis of the ellipse such that tan $\left({\mathrm{\theta }}_1 + {\mathrm{\theta }}_2\right) = \mathrm{k}$ constant. The locus of the pt. of intersection of the tangents $({\mathrm{x}}^2 – {\mathrm{a}}^2) – ({\mathrm{y}}^2 – {\mathrm{b}}^2) =$ 

$$\begin{gathered} \mathrm{Let} \mathrm{y}=\mathrm{mx}\pm \sqrt{{\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2} \mathrm{be} \mathrm{tangent} \mathrm{drawn} \mathrm{through} \mathrm{the} \mathrm{point} ({\mathrm{x}}_1,{\mathrm{y}}_1) \\ \Rightarrow {\mathrm{m}}^2({{\mathrm{x}}_1}^2-{\mathrm{a}}^2)-2{\mathrm{mx}}_1{\mathrm{y}}_1+{{\mathrm{y}}_1}^2-{{\mathrm{b}}^2}_{ }=0 \\ \mathrm{Let} {\mathrm{m}}_1={\mathrm{Tan\theta }}_1 {\mathrm{m}}_2={\mathrm{Tan\theta }}_2 \\ {\mathrm{m}}_1+{\mathrm{m}}_2=\frac{2{\mathrm{x}}_1{\mathrm{y}}_1}{{{\mathrm{x}}_1}^2-{\mathrm{a}}^2} \\ {\mathrm{m}}_1{\mathrm{m}}_2=\frac{{{\mathrm{y}}_1}^2-{\mathrm{b}}^2}{{{\mathrm{x}}_1}^2-{\mathrm{a}}^2} \\ \mathrm{Tan}({\mathrm{\theta }}_1+{\mathrm{\theta }}_2)=\mathrm{k} \\ \frac{{\mathrm{m}}_1+{\mathrm{m}}_2}{1-{\mathrm{m}}_1{\mathrm{m}}_2}=\mathrm{k} \\ \frac{{\displaystyle \frac{2{\mathrm{x}}_1{\mathrm{y}}_1}{{{\mathrm{x}}_1}^2-{\mathrm{a}}^2}}}{1-{\displaystyle \frac{{{\mathrm{y}}_1}^2-{\mathrm{b}}^2}{{{\mathrm{x}}_1}^2-{\mathrm{a}}^2}}}=\mathrm{k} \\ \frac{2{\mathrm{x}}_1{\mathrm{y}}_1}{{{\mathrm{x}}_1}^2-{\mathrm{a}}^2-{{\mathrm{y}}_1}^2+{\mathrm{b}}^2}=\mathrm{k} \\ \mathrm{locus} \mathrm{of} ({\mathrm{x}}_1,{\mathrm{y}}_1) \mathrm{is} \\ 2\mathrm{xy}=\mathrm{k}\left({\mathrm{x}}^2-{\mathrm{y}}^2-{\mathrm{a}}^2+{\mathrm{b}}^2\right) \end{gathered}$$