The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

In an ellipse, the distance between its foci is $6$ and minor axis is $8.$ Then its eccentricity is :

The equation of the ellipse whose one of the vertices is $(0,7)$ and the corresponding directrix is $y = 12$, is

Let $\mathrm{E}$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus at $(4,-4)$ and one vertex at $(5,-4) .$ If $m x-y=4, m\,>\,0$ is a tangent to the ellipse $\mathrm{E}$, then the value of $5 \mathrm{~m}^{2}$ is equal to $.....$

Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Then the square of the slope of the line $L$ is ..... .

The locus of the poles of normal chords of an ellipse is given by