If $P$ is a point on $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with foci $S$ and $S^{\prime}$,then the maximum area of $\triangle S P S^{\prime}$ is

The center of the ellipse $x^2+2y^2-4x+12y+14=0$ is

If the foci and vertices of an ellipse are $(\pm 1, 0)$ and $(\pm 2, 0)$ respectively,then the length of the minor axis of the ellipse is:

The foci of the ellipse $25x^2 + 4y^2 + 100x - 4y + 100 = 0$ are

Let $x = 9$ be a directrix of an ellipse $E$,whose centre is at the origin and eccentricity is $1/3$. Let $P(\alpha, 0), \alpha > 0$,be a focus of $E$ and $AB$ be a chord passing through $P$. Then the locus of the mid point of $AB$ is :

Consider an ellipse,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