The locus of the foot of the perpendicular drawn from the centre of the ellipse $x^2 + 3y^2 = 6$ to any tangent to it is:

The eccentricity of an ellipse,with its centre at the origin,is $\frac{1}{2}$. If one of the directrices is $x = 4$,then the equation of the ellipse 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:

Find the equation for the ellipse that satisfies the given conditions: Length of minor axis $16$,foci $(0, \pm 6)$.

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 line $2x + 5y = 12$ intersects the ellipse $4x^2 + 5y^2 = 20$ in two distinct points $A$ and $B$,then the mid-point of $AB$ is