Find the condition for the line $\mathrm{xcos}\mathrm{\alpha }+\mathrm{ysin}\mathrm{\alpha }=\mathrm{p}$ to be a tangent to ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

Given ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Given line $\mathrm{xcos}\mathrm{\alpha }+\mathrm{ysin}\mathrm{\alpha }=\mathrm{p}$
$\mathrm{ysin}\mathrm{\alpha }=-\mathrm{xcos}\mathrm{\alpha }+\mathrm{p}$
$\mathrm{y}=-\frac{\cos \mathrm{\alpha }}{\sin \mathrm{\alpha }}\mathrm{x}+\frac{\mathrm{p}}{\sin \mathrm{\alpha }}$ is a tangent to ellipse 

$\begin{array}{l}\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1 then\\ {\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2\\ \frac{{\mathrm{p}}^2}{{\sin }^2\mathrm{\alpha }}={\mathrm{a}}^2\cdot \frac{{\cos }^2\mathrm{\alpha }}{{\sin }^2\mathrm{\alpha }}+{\mathrm{b}}^2\\ {\mathrm{p}}^2={\mathrm{a}}^2{\cos }^2\mathrm{\alpha }+{\mathrm{b}}^2{\sin }^2\mathrm{\alpha }\end{array}$