$\text{ If the tangent at any point of an ellipse }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1\text{, where }\mathrm{a}>\mathrm{b}\text{, makes angle of }$

$\text{ inclinations }\mathrm{\alpha }=\frac{5\mathrm{\pi }}{6}\text{, with the major axis and angle }\mathrm{\beta }=\frac{\mathrm{\pi }}{3}\text{ with the focal radius of }$

$\text{ the point of contact then the eccentricity e of the ellipse is }$

If any tangent to any point of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a>b$ makes angle of inclinations $\alpha$with the major axis and the an angle $\beta$with the focal radius of the point of contact then the eccentricity of the ellipse is $\frac{\cos \beta }{\cos \alpha }$

Hence the Eccentricity of the given ellipse is $\frac{1}{\sqrt{3}}$