At which point on the ellipse $4x^2 + 9y^2 = 1$ is the tangent parallel to $8x = 9y$?

If $P$ is any point on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ and $S$ and $S^{\prime}$ are the foci,then $PS + PS^{\prime}$ is equal to

If the normal at the point $P(\theta)$ to the ellipse $\frac{x^2}{14} + \frac{y^2}{5} = 1$ intersects it again at the point $Q(2\theta)$,then $\cos \theta$ is equal to

Let $P$ be a variable point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with foci $F_1$ and $F_2$. If $A$ is the area of triangle $PF_1F_2$,then the maximum value of $A$ is:

The equation of an ellipse in its standard form,given the distance between its foci is $2$ units and the length of its latus rectum is $\frac{15}{2}$ units,is

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: