$P$ is the extremity of the latus rectum of the ellipse $3x^{2} + 4y^{2} = 48$ in the first quadrant. The eccentric angle of $P$ is

If the tangents drawn from a point $P$ to the ellipse $4x^2+9y^2-16x+54y+61=0$ are perpendicular,then the locus of $P$ is

Let a line $L$ pass through the point of intersection of the lines $bx + 10y - 8 = 0$ and $2x - 3y = 0$,where $b \in R - \{\frac{4}{3}\}$. If the line $L$ also passes through the point $(1, 1)$ and touches the circle $17(x^2 + y^2) = 16$,then the eccentricity of the ellipse $\frac{x^2}{5} + \frac{y^2}{b^2} = 1$ is:

Let $S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0$ and $S^{\prime} \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0$ be two intersecting ellipses. If $P(a \cos \theta, b \sin \theta)$ and $Q\left(a \cos \left(\frac{\pi}{2}+\theta\right), b \sin \left(\frac{\pi}{2}+\theta\right)\right)$ are their points of intersection,then $\frac{1}{2}\left(a^2 \beta^2+b^2 \alpha^2\right)=$

Which of the following points lies on the locus of the foot of the perpendicular drawn from any of the foci of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ to any of its tangents?

The equations of the tangent and normal at point $(3, -2)$ of the ellipse $4x^2 + 9y^2 = 36$ are: