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 locus of the point of intersection of the perpendicular tangents to the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ is

The length of the minor axis (along $y-$axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}} .$ If this ellipse touches the line, $x+6 y=8 ;$ then its eccentricity is

If two tangents drawn from a point $(\alpha, \beta)$ lying on the ellipse $25 x^{2}+4 y^{2}=1$ to the parabola $y^{2}=4 x$ are such that the slope of one tangent is four times the other, then the value of $(10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}$ equals

If the eccentricity of the two ellipse $\frac{{{x^2}}}{{169}} + \frac{{{y^2}}}{{25}} = 1$ and $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ are equal, then the value of $a/b$ is

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2,3)$