Ellipse

Find the equation of the ellipse having its centre at the point $C\left(2,-3\right)$, one focus at $S\left(3,-3\right)$ and one vertex at $A\left(4,-3\right)$.

If the angle between the straight lines joining the foci to an extremity of minor axis in an ellipse be $90°$, then the eccentricity of the ellipse is

The eccentricity of the ellipse, which meets the straight line $\frac{x}{7}+\frac{y}{2}=1$ on the axis of $x$ and the straight line $\frac{x}{3}-\frac{y}{5}=1$ on the axis of $y$ and whose axes lie along the axes of coordinates, is

A variable point $P$ on the ellipse of eccentricity $e$ is joined to the foci $S$ and $S\text{'}$. The locus of the in-centre of the triangle $PSS\text{'}$ is an ellipse whose eccentricity is

The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus rectum is

If latus rectum of an ellipse is half of its minor axis, then its eccentricity is

If $C$ is the centre and $A, B$ are the two points on the conic $4x^2+9y^2-8x-36y+4=0$ such that $\angle ACB=\frac{\pi }{2},$ then $C{A}^{-2}+C{B}^{-2}$ is equal to

The points of intersection of the two ellipses $x^2+2y^2-6x-12y+23=0, 4x^2+2y^2-20x-12y+35=0$

If $P\left(\theta \right)$ and $Q\left(\frac{\pi }{2}+\theta \right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then locus of the mid-point of $PQ$ is

The foci of an ellipse are $(\pm 2,0)$ and its eccentricity is $\frac{1}{2}$, find its equation.