The equation of the auxiliary circle of the ellipse $16 x^{2}+25 y^{2}+32 x-100 y=284$ is

In an ellipse,the distance between its foci is $6$ and the minor axis is $8$. Then its eccentricity is:

The eccentricity of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ is

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

$A$ rod $AB$ of length $15 \ cm$ rests between two coordinate axes such that the end point $A$ lies on the $x-$axis and the end point $B$ lies on the $y-$axis. $A$ point $P(x, y)$ is taken on the rod such that $AP = 6 \ cm$. Show that the locus of $P$ is an ellipse.

If any tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ cuts off intercepts of length $h$ and $k$ on the axes,then $\frac{a^2}{h^2} + \frac{b^2}{k^2} = $