If the tangents drawn from a point $P$ to the ellipse $4 x^2+9 y^2-24 x+36 y=0$ are perpendicular,then the locus of $P$ 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

Let $A, B,$ and $C$ be three points on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$. The line joining $A$ and $C$ is parallel to the $x$-axis,and $B$ is the endpoint of the minor axis whose ordinate is positive. Find the maximum area of $\Delta ABC$.

Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$,let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$,and $M(P, Q^{\prime})$ be the mid-point of the line segment joining $P$ and $Q^{\prime}$. Then the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q^{\prime})$,as $P, Q$ and $Q^{\prime}$ vary on $E$,is:

The ratio of the distance of a point from a fixed point and a line $x = 9/2$ is always $2 : 3$. Then the locus of the point is

In an ellipse,the distance between its foci is $6$ and the length of the major axis is $8$. Find its eccentricity.