If the distance between a focus and the corresponding directrix of an ellipse is $8$ and the eccentricity is $1/2$,then the length of the minor axis is

Let a tangent drawn at any point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ cut the $X$-axis at $Q$. Let $R$ be the image of $Q$ with respect to $y=x$. If $S$ is a circle with $QR$ as its diameter,then the fixed point through which the circle $S$ passes is

The equation of the ellipse whose one focus is at $(4, 0)$ and whose eccentricity is $4/5$,is

If $F_1$ and $F_2$ are the feet of the perpendiculars from the foci $S_1$ and $S_2$ of an ellipse $\frac{x^2}{5} + \frac{y^2}{3} = 1$ on the tangent at any point $P$ on the ellipse,then $(S_1 F_1) (S_2 F_2)$ is equal to

Let $E$ be an ellipse whose major axis is the $X$-axis and minor axis is the $Y$-axis. If the distances of a point $P \left(\frac{5}{2}, 2 \sqrt{3}\right)$ on $E$ from its foci are $\frac{7}{2}$ and $\frac{13}{2}$,then the eccentricity of the ellipse $E$ is

If the product of the lengths of the perpendiculars drawn from the foci to the tangent $y = \frac{-3}{4}x + 3\sqrt{2}$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $9$,then the eccentricity of that ellipse is