Inscribed triangle in an ellipse

Question

An ellipse has eccentricity $0.8$ and a line bisecting the semi-minor axis $AB$ perpendicularly cutting the ellipse at point $O$. Find angle $\hat{AOB}$.

Answer 1

If the ellipse has equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then line containing $AB$ has equation $\frac xa+\frac yb=1$, hence $\tan(\alpha)=\frac ba$ where $\alpha=\frac{\hat{AOB}}2$. Since the eccentricity $e^2=1-\frac{b^2}{a^2}=1-\tan^2(\alpha)$ we get $\tan(\alpha)=\sqrt{1-e^2}$, hence $\hat{AOB}=2\arctan\sqrt{1-e^2}$.