Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$-axis. If its minor axis subtends an angle $60^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to $...........$.

If $x^{2}+9 y^{2}-4 x+3=0, x, y \in R$, then $x$ and $y$ respectively lie in the intervals:

The radius of the circle having its centre at $(0, 3)$ and passing through the foci of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1$, is

If the tangent at a point on the ellipse $\frac{{{x^2}}}{{27}} + \frac{{{y^2}}}{3} = 1$ meets the coordinate axes at $A$ and $B,$  and  $O$  is the origin, then the minimum area (in sq. units) of the triangle $OAB$  is

Let $F_1$ & $F_2$ be the foci of an ellipse $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} = 1$ such that a ray from $F_1$ strikes the elliptical mirror at the point $P$ and get reflected. Then equation of angle bisector of the angle between incident ray and reflected ray can be 

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ from any of its foci?