The length of the latus-rectum of the ellipse,whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$,is

The tangents at the extremities of the latus rectum of the ellipse $3x^2 + 4y^2 = 12$ form a rhombus of area (in $sq. \ units$):

If $OB$ is the semi-minor axis of an ellipse,$F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle,then the square of the eccentricity of the ellipse is

The locus of the point of intersection of two perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is:

The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line $x+6y=8$,then its eccentricity is

$A$ tangent drawn at a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ cuts the $X$-axis at point $A$. If $A^{\prime}$ is the image of $A$ with respect to the line $y=x$,then the circle with $AA^{\prime}$ as its diameter passes through the fixed point: