Area (in sq. units) of the region outside $\frac{|\mathrm{x}|}{2}+\frac{|\mathrm{y}|}{3}=1$ and inside the ellipse $\frac{\mathrm{x}^{2}}{4}+\frac{\mathrm{y}^{2}}{9}=1$ is

The acute angle between the pair of tangents drawn to the ellipse $2 x^{2}+3 y^{2}=5$ from the point $(1,3)$ is.

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse ${x^2} + 2{y^2} = 2$ between the co-ordinates axes, is

The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

Planet $M$ orbits around its sun, $S$, in an elliptical orbit with the sun at one of the foci. When $M$ is closest to $S$, it is $2\,unit$ away. When $M$ is farthest from $S$, it is $18\, unit$ away, then the equation of motion of planet $M$ around its sun $S$, assuming $S$ at the centre of the coordinate plane and the other focus lie on negative $y-$ axis, is