A tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ intersect the co-ordinate axes at $A$ and $B,$ then locus of circumcentre of triangle $AOB$ (where $O$ is origin) is

Find the equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\,\pm 5)$

Let the common tangents to the curves $4\left(x^{2}+y^{2}\right)=$ $9$ and $y ^{2}=4 x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and $6$ , respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac{l}{ e ^{2}}$ is equal to

Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Then the square of the slope of the line $L$ is ..... .

The position of the point $(1, 3)$ with respect to the ellipse $4{x^2} + 9{y^2} - 16x - 54y + 61 = 0$