If $l$ and $b$ are respectively the length and breadth of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$,then $(l, b) =$

The locus of the point of intersection of two perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is:

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ such that $y_1 > 0$ and $y_2 > 0$. Let $C$ denote the circle $x^2+y^2=9$,and $M$ be the point $(3,0)$. Suppose the line $x=x_1$ intersects $C$ at $R$,and the line $x=x_2$ intersects $C$ at $S$,such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle ROM = \frac{\pi}{6}$ and $\angle SOM = \frac{\pi}{3}$,where $O$ denotes the origin $(0,0)$. Let $|XY|$ denote the length of the line segment $XY$. Then which of the following statements is (are) True?
$(A)$ The equation of the line joining $P$ and $Q$ is $2x+3y=3(1+\sqrt{3})$
$(B)$ The equation of the line joining $P$ and $Q$ is $2x+y=3(1+\sqrt{3})$
$(C)$ If $N_2=(x_2, 0)$,then $3|N_2Q|=2|N_2S|$
$(D)$ If $N_1=(x_1, 0)$,then $9|N_1P|=4|N_1R|$

If the coordinates of two points $A$ and $B$ are $(\sqrt{7}, 0)$ and $(-\sqrt{7}, 0)$ respectively and $P$ is any point on the conic $9x^{2} + 16y^{2} = 144$,then $PA + PB$ is equal to

If the equations $x = 1 + 2 \cos \theta$ and $y = 2 + \sin \theta$ for $0 \leq \theta < 2 \pi$ represent an ellipse,then the point of intersection of the normal drawn at $P(\theta = \pi/4)$ to this ellipse and its major axis is:

$A$ line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{x^2}{36} + \frac{y^2}{25} = 1$ at $A$ and $B$ such that $(PA) \cdot (PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to: