If $m$ is the slope of a common tangent to the curves $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and $x^{2}+y^{2}=12$, then $12\; m ^{2}$ is equal to

The equation of an ellipse whose focus $(-1, 1)$, whose directrix is $x - y + 3 = 0$ and whose eccentricity is $\frac{1}{2}$, is given by

Let $P(2,2)$ be a point on an ellipse whose foci are $(5,2)$ and $(2,6)$, then eccentricity of ellipse is 

Number of tangents to the circle $x^2 + y^2 = 3$ , which are normal to the ellipse $4x^2 + 9y^2 = 36$ , is

For an ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ with vertices $A$  and $ A', $ tangent drawn at the point $P$  in the first quadrant meets the $y-$axis in $Q $ and the chord $ A'P$ meets the $y-$axis in $M.$  If $ 'O' $ is the origin then $OQ^2 - MQ^2$  equals to

Let the line $y=m x$ and the ellipse $2 x^{2}+y^{2}=1$ intersect at a ponit $\mathrm{P}$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $\left(-\frac{1}{3 \sqrt{2}}, 0\right)$ and $(0, \beta),$ then $\beta$ is equal to