P.T the condition for the straight line y = mx + c to be a tangent to the Ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\mathrm{a}>\mathrm{b})\text{ is }{\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2$

Suppose y = mx + c ... (1)  is a tangent to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1\text{ at }\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$
The equation of the tangent at P is  $\frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}+\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}-1=0 ...\left(2\right)$
(1) & (2) represent the same line
$\begin{array}{l}\therefore \frac{{\mathrm{x}}_1}{{\mathrm{a}}^2\mathrm{m}}=\frac{{\mathrm{y}}_1}{{\mathrm{b}}^2(-1)}=\frac{-1}{\mathrm{c}}\\ \Rightarrow {\mathrm{x}}_1=\frac{-{\mathrm{a}}^2\mathrm{m}}{\mathrm{c}},{\mathrm{y}}_1=\frac{{\mathrm{b}}^2}{\mathrm{c}}\\ \Rightarrow \mathrm{P}=\left(\frac{-{\mathrm{a}}^2\mathrm{m}}{\mathrm{c}},\frac{{\mathrm{b}}^2}{\mathrm{c}}\right)\end{array}$
Since P lies on the line y = mx + c then 
$\begin{array}{l}\Rightarrow {\mathrm{y}}_1={\mathrm{mx}}_1+\mathrm{c}\Rightarrow \frac{{\mathrm{b}}^2}{\mathrm{c}}=\mathrm{m}\left(\frac{-{\mathrm{a}}^2\mathrm{m}}{\mathrm{c}}\right)+\mathrm{c}\\ \Rightarrow {\mathrm{b}}^2=-{\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{c}}^2\Rightarrow {\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2\end{array}$