An ellipse having foci at $(3, 1)$ and $(1, 1)$ passes through the point $(1, 3)$. Then its eccentricity is

The locus of the mid-points of the chords of an ellipse $x^{2}+4y^{2}=4$ that are drawn from the positive end of the minor axis is

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 :

If the line $2x + 5y = 12$ intersects the ellipse $4x^2 + 5y^2 = 20$ in two distinct points $A$ and $B$,then the mid-point of $AB$ is

If the straight line $8x + 3\sqrt{2}y = 36$ touches the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 2$ at $(a, b)$,then $a + \sqrt{2}b =$

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 $\Delta ABC$.