If a tangent to the ellipse $x^{2}+4 y^{2}=4$ meets the tangents at the extremities of its major axis at $\mathrm{B}$ and $\mathrm{C}$, then the circle with $\mathrm{BC}$ as diameter passes through the point:

The eccentricity of the ellipse $9{x^2} + 5{y^2} - 30y = 0$, is

The equations of the common tangents to the ellipse, $ x^2 + 4y^2 = 8 $ $\&$  the parabola $y^2 = 4x$  can be

Consider ellipses $E _{ k }: kx ^2+ k ^2 y ^2=1, k =1,2, \ldots$,$20$. Let $C _{ k }$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$, If $r_k$ is the radius of the circle $C _{ k }$, then the value of $\sum \limits_{ k =1}^{20} \frac{1}{ I _{ k }^2}$ is $.......$.

The lengths of major and minor axis of an ellipse are $10$ and $8$ respectively and its major axis along the $y$ - axis. The equation of the ellipse referred to its centre as origin is

Eccentric angle of a point on the ellipse ${x^2} + 3{y^2} = 6$ at a distance $2$ units from the centre of the ellipse is