The locus of the foot of perpendicular drawn from the centre of the ellipse ${x^2} + 3{y^2} = 6$ on any tangent to it is

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

Length of common chord of the ellipse ${\frac{{\left( {x - 2} \right)}}{9}^2} + {\frac{{\left( {y + 2} \right)}}{4}^2} = 1$ and the circle ${x^2} + {y^2} - 4x + 2y + 4 = 0$

If lines $3x + 2y = 10$ and $-3x + 2y = 10$ are tangents at the extremities of latus rectum of an ellipse whose centre is origin, then the length of latus rectum of ellipse is 

The distance between the directrices of the ellipse $\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{20}} = 1$ is

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