If a tangent to the ellipse  $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\mathrm{a}>\mathrm{b})$ meets major and minor axis at M and N respectivly and C is the centre of the ellipse  then $\frac{{\mathrm{a}}^2}{{\left(\mathrm{CM}\right)}^2}+\frac{{\mathrm{b}}^2}{{\left(\mathrm{CN}\right)}^2}=$

 

$\mathrm{eq}uation \mathrm{of} t\mathrm{angent} \mathrm{is} \frac{\mathrm{x}}{\mathrm{a}}\mathrm{cos\theta }+\frac{\mathrm{y}}{\mathrm{b}}\mathrm{sin\theta }=1 \mathrm{meets} \mathrm{coordinate} \mathrm{axes} at \mathrm{M} \& \mathrm{N}$

$$\begin{gathered} \therefore \mathrm{M}=\left(\frac{\mathrm{a}}{\mathrm{cos\theta }}, 0\right) ,\mathrm{N}\left(0, \frac{\mathrm{b}}{\mathrm{sin\theta }}\right) \\ \frac{{\mathrm{a}}^2}{{\mathrm{CM}}^2}+\frac{{\mathrm{b}}^2}{{\mathrm{CN}}^2}=\frac{{\mathrm{a}}^2}{{\displaystyle \frac{{\mathrm{a}}^2}{{\cos }^2\mathrm{\theta }}}}+\frac{{\mathrm{b}}^2}{{\displaystyle \frac{{\mathrm{b}}^2}{{\sin }^2\mathrm{\theta }}}} \\ ={\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta } \\ =1 \end{gathered}$$