The equations of the common tangents to the ellipse, $ x^2 + 4y^2 = 8 $ $\&$  the parabola $y^2 = 4x$  can be

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

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is

The centre of an ellipse is $C$ and $PN$ is any ordinate and $A$, $A’$ are the end points of major axis, then the value of $\frac{{P{N^2}}}{{AN\;.\;A'N}}$ is

Let a tangent to the Curve $9 x^2+16 y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $A B$ is $.........$

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^2+9 y^2=9$ meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A, M$ and the origin $O$ is