The equation of the ellipse whose centre is at origin and which passes through the points $(-3, 1)$ and $(2, -2)$ is

If any tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ cuts off intercepts of length $h$ and $k$ on the axes, then $\frac{{{a^2}}}{{{h^2}}} + \frac{{{b^2}}}{{{k^2}}} = $

A tangent is drawn to the ellipse $\frac{{{x^2}}}{{32}} + \frac{{{y^2}}}{8} = 1$ from the point $A(8, 0)$ to touch the ellipse at point $P.$ If the normal at $P$ meets the major axis of ellipse at point $B,$ then the length $BC$ is equal to (where $C$ is centre of ellipse) - ............ $\mathrm{units}$

Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ are $\left(f_1, 0\right)$ and $\left(f_2, 0\right)$ where $f_1>0$ and $f_2<0$. Let $P _1$ and $P _2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $\left(f_1, 0\right)$ and $\left(2 f_2, 0\right)$, respectively. Let $T_1$ be a tangent to $P_1$ which passes through $\left(2 f_2, 0\right)$ and $T_2$ be a tangent to $P_2$ which passes through $\left(f_1, 0\right)$. The $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$, then the value of $\left(\frac{1}{m^2}+m_2^2\right)$ is

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

If $C$ is the centre of the ellipse $9x^2 + 16y^2$ = $144$ and $S$ is one focus. The ratio of $CS$ to major axis, is