For what values of $t$ does the line $x = at^2$ intersect the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at real points?

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:

The sum of focal distances of any point on the ellipse with major and minor axes as $2a$ and $2b$ respectively,is equal to

The locus of the point of intersection of perpendicular tangents to the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ is:

For an ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ with vertices $A$ and $A'$,a tangent drawn at the point $P$ in the first quadrant meets the $y$-axis at $Q$,and the chord $A'P$ meets the $y$-axis at $M$. If $O$ is the origin,then $OQ^2 - MQ^2$ is equal to:

The eccentric angle of the point $(2, \sqrt{3})$ lying on $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ is