Number of points on the ellipse $\frac{x^2}{50} + \frac{y^2}{20} = 1$ from which a pair of perpendicular tangents are drawn to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is:

The locus of the point of intersection of mutually perpendicular tangents to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

The equation of the tangent to the ellipse $x^2 + 16y^2 = 16$ making an angle of $60^\circ$ with the $x$-axis is

If the latus rectum of an ellipse is equal to half of its minor axis,then its eccentricity is

If $\alpha$ and $\beta$ are the eccentric angles of the endpoints of a focal chord of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2} = ....$

Let $E_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a > b$. Let $E_{2}$ be another ellipse such that it touches the end points of the major axis of $E_{1}$ and the foci of $E_{2}$ are the end points of the minor axis of $E_{1}$. If $E_{1}$ and $E_{2}$ have the same eccentricity $e$,then the value of $e$ is: