The length of the latus rectum of the ellipse $4x^2 + 9y^2 - 8x - 36y + 4 = 0$ is

Find the locus of a point such that the sum of its distances from the points $(0, 2)$ and $(0, -2)$ is $6$.

On the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$,let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x+2y=0$. Let $S$ and $S'$ be the foci of the ellipse and $e$ be its eccentricity. If $A$ is the area of the triangle $SPS'$,then the value of $(5-e^{2}) \cdot A$ is:

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(0, \pm \sqrt{5})$,ends of minor axis $(\pm 1, 0)$.

In an ellipse,two vertices are $(5,0)$ and $(0,-4)$. Then the equation of the ellipse is

$P$ is a point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. When the area of $\Delta PSS'$ is maximum,the inradius of $\Delta PSS'$ ($S$ and $S'$ are foci) is equal to: