The parametric equations of the ellipse whose foci are $(-3, 0)$ and $(9, 0)$ and eccentricity is $\frac{1}{3}$,are

If lines $3x + 2y = 10$ and $-3x + 2y = 10$ are tangents at the extremities of the latus rectum of an ellipse whose centre is the origin,then the length of the latus rectum of the ellipse is:

$A$ vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta(h)=$ area of the triangle $PQR$,$\Delta_1=\max _{1 / 2 \leq h \leq 1} \Delta(h)$ and $\Delta_2=\min _{1 / 2 \leq h \leq 1} \Delta(h)$,then $\frac{8}{\sqrt{5}} \Delta_1-8 \Delta_2=$

Consider the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

The line $lx + my + n = 0$ will cut the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at points whose eccentric angles differ by $\pi/2$ if:

Consider the conic $e x^2 + \pi y^2 - 2 e^2 x - 2 \pi^2 y + e^3 + \pi^3 = \pi e$.
Suppose $P$ is any point on the conic and $S_1, S_2$ are the foci of the conic,then the maximum value of $(P S_1 + P S_2)$ is