A circle of radius 4, is concentric with the ellipse 3x² + 13y² = 78. Prove that a common tangent is inclined to the major axis at angle $\frac{\mathrm{\pi }}{4}$.

Equation of the ellipse 3x² + 13y² = 78
$\frac{{\mathrm{x}}^2}{26}+\frac{{\mathrm{y}}^2}{6}=1$ ...(1)
Centre of the ellipse is (0, 0)
equation of the circle is ${\mathrm{x}}^2+{\mathrm{y}}^2=16 ...\left(2\right)$
Equation of tangent at p($\mathrm{\theta }$)  to the circle is
$\begin{array}{l}\mathrm{xcos}\mathrm{\theta }+\mathrm{ysin}\mathrm{\theta }=4\\ \mathrm{y}=\frac{-\cos \mathrm{\theta }}{\sin \mathrm{\theta }}\cdot \mathrm{x}+\frac{4}{\sin \mathrm{\theta }}\end{array}$
$\text{ (3) is tangent to the ellipse then }{\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2$
$\begin{array}{l}\Rightarrow \frac{16}{{\sin }^2\mathrm{\theta }}=26\cdot \frac{{\cos }^2\mathrm{\theta }}{{\sin }^2\mathrm{\theta }}+6\\ \Rightarrow 16=26{\cos }^2\mathrm{\theta }+6{\sin }^2\mathrm{\theta }\\ \Rightarrow 16=26{\cos }^2\mathrm{\theta }+6\cdot \left(1-{\cos }^2\mathrm{\theta }\right)\\ \Rightarrow 16=26{\cos }^2\mathrm{\theta }+6-6{\cos }^2\mathrm{\theta }\\ \Rightarrow 20{\cos }^2\mathrm{\theta }=10\Rightarrow {\cos }^2\mathrm{\theta }=\frac{10}{20}=\frac{1}{2}\\ \Rightarrow \cos \mathrm{\theta }=\frac{1}{\sqrt{2}}\Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{4}\end{array}$
$\Rightarrow \text{ Angle made by tangent with major axis is }\frac{\mathrm{\pi }}{4}$