The locus of the point of intersection of two tangents to the ellipse $x^2/a^2+y^2/b^2=1$ which make complementary angles with the major axis of the ellipse is 

If $P\left(x_1,y_1\right)$ is the point of intersection of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with slopes $m_1$  and  $m_2\text{ then }{m}_1{m}_2=\frac{y_1^2-b^2}{x_1^2-a^2}\text{ where }{m}_1=\tan {\theta }_1\mathrm{\&}\tan {\theta }_2$

Since  ${\theta }_1,{\theta }_2$ are complementary angles, we have $\tan {\theta }_1\tan {\theta }_2=1$

$\Rightarrow \frac{y_1^2-b^2}{x_1^2-a^2}=1\Rightarrow x_1^2-y_1^2=a^2-b^2.\therefore \text{ Locus of }P\left(x_1,y_1\right)\text{ is }{x}^2-y^2=a^2-b^2$