Let $P$ be an arbitrary point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $a > b > 0$. Suppose $F_1$ and $F_2$ are the foci of the ellipse. The locus of the centroid of the $\Delta P F_1 F_2$ as $P$ moves on the ellipse is

The equation of the tangents drawn at the ends of the major axis of the ellipse $9{x^2} + 5{y^2} - 30y = 0$, are

Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 6,\,0),$ foci $(\pm 4,\,0)$

The equation of the tangent at the point $(1/4, 1/4)$ of the ellipse $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{{12}} = 1$ is

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  the circle passing through the foci of the $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1$ and having centre at $(0,3) $ is