The line $x=m^2$ meets the ellipse $9x^2+y^2=9$ at real and distinct points if and only if

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 $2$ units away. When $M$ is farthest from $S$,it is $18$ units away. Assuming $S$ is at the origin $(0, 0)$ and the other focus lies on the negative $y$-axis,find the equation of the elliptical orbit of planet $M$.

The area enclosed by the ellipse $4x^2 + 9y^2 = 1$ is . . . . . . sq. units.

If the normal at one end of a latus rectum of the ellipse $\frac{x^2}{32}+\frac{y^2}{b^2}=1$ passes through one end of the minor axis,then $\frac{e^4}{1-e^2}=$ (Here $e$ is the eccentricity of the ellipse)

If $P \equiv (x, y)$,$F_1 \equiv (3, 0)$,$F_2 \equiv (-3, 0)$ and $16x^2 + 25y^2 = 400$,then $PF_1 + PF_2$ equals

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$,then the length of its latus rectum is