The product of the lengths of perpendiculars from the foci on any tangent to the ellipse $3x^2 + 5y^2 = 1$ is:

The equations of the latus rectum of the ellipse $9x^2+4y^2-18x-8y-23=0$ are

Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$,let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$,and $M(P, Q^{\prime})$ be the mid-point of the line segment joining $P$ and $Q^{\prime}$. Then the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q^{\prime})$,as $P, Q$ and $Q^{\prime}$ vary on $E$,is:

Find the equation of the ellipse whose axes are the coordinate axes,which passes through the point $(-3, 1)$ and has an eccentricity of $\sqrt{2/5}$.

The values of $c$ such that the line $y=4x+c$ touches the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are

The eccentricity of an ellipse passing through $(3 \sqrt{2}, \sqrt{10})$ with foci at $(-4,0)$ and $(4,0)$ is