Tangents at extremities of latus rectum of ellipse $3x^2 + 4y^2 = 12$ form a rhombus of area (in $sq.\ units$) -

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$

The angle of intersection of ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and circle ${x^2} + {y^2} = ab$, is

The equation of the ellipse whose latus rectum is $8$ and whose eccentricity is $\frac{1}{{\sqrt 2 }}$, referred to the principal axes of coordinates, is

The ellipse $ 4x^2 + 9y^2 = 36$  and the hyperbola $ 4x^2 -y^2 = 4$  have the same foci and they intersect at right angles then the equation of the circle through the points of intersection of two conics 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