The normal at a variable point $P$ on an ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}}= 1$  of eccentricity e meets the axes of the ellipse in $ Q$  and $R$  then the locus of the mid-point of $QR$  is a conic with an eccentricity $e' $  such that :

Tangents are drawn from points onthe circle $x^2 + y^2 = 49$ to the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{24}} = 1$ angle between the tangents is

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is $10$ and one of the foci is at $(0, 5\sqrt 3 )$, then the length of its latus rectum is

In the ellipse, minor axis is $8$ and eccentricity is $\frac{{\sqrt 5 }}{3}$. Then major axis is

The latus rectum of an ellipse is $10$ and the minor axis is equal to the distance between the foci. The equation of the ellipse is

Consider an elIipse, whose centre is at the origin and its major axis is along the $x-$ axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is $6$, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is