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

Let $P \left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), Q , R$ and $S$ be four points on the ellipse $9 x^2+4 y^2=36$. Let $P Q$ and $RS$ be mutually perpendicular and pass through the origin. If $\frac{1}{( PQ )^2}+\frac{1}{( RS )^2}=\frac{ p }{ q }$, where $p$ and $q$ are coprime, then $p+q$ is equal to $………$.

The line $x\cos \alpha + y\sin \alpha = p$ will be a tangent to the conic $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, if

Two sets $A$ and $B$ are as under:

$A = \{ \left( {a,b} \right) \in R \times R:\left| {a – 5} \right| < 1 \,\,and\,\,\left| {b – 5} \right| < 1\} $; $B = \left\{ {\left( {a,b} \right) \in R \times R:4{{\left( {a – 6} \right)}^2} + 9{{\left( {b – 5} \right)}^2} \le 36} \right\}$ then : . . . . .

If $F_1$ and $F_2$ be the feet of the perpendicular 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