The centre of the ellipse $4x^2 + 9y^2 - 16x - 54y + 61 = 0$ is

If the line $x \cos \alpha + y \sin \alpha = p$ is normal to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,then

The eccentric angle of a point on the ellipse $x^2 + 3y^2 = 6$ at a distance of $2$ units from the centre of the ellipse is:

$P$ is a point on the conic $a^2 x^2+b^2 y^2=a^2(a^2+b^2-y^2)$ and $S$ is a focus of that conic. $M$ is the foot of the perpendicular from $P$ onto a directrix of that conic nearer to $S$. If $PM = K SP$,then $K=$

Tangent to the ellipse $\frac{x^{2}}{32}+\frac{y^{2}}{18}=1$ having slope $-\frac{3}{4}$ meets the coordinate axes at $A$ and $B$. Find the area of the $\Delta AOB$,where $O$ is the origin.

The locus of the point of intersection of mutually perpendicular tangents to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is