The length of the latus rectum of $x^2+3y^2=12$ is

$B$ is an extremity of the minor axis of an ellipse whose foci are $S$ and $S^{\prime}$. If $\angle SBS^{\prime}$ is a right angle,then the eccentricity of the ellipse is

The foci of the ellipse $25(x + 1)^2 + 9(y + 2)^2 = 225$ are at

If the distance between the directrices is thrice the distance between the foci,then the eccentricity of the ellipse 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 $2$ 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 line $x=m^2$ meets the ellipse $9x^2+y^2=9$ at real and distinct points if and only if