The equation of the normal at the point $(2, 3)$ on the ellipse $9x^2 + 16y^2 = 180$ is:

The midpoint of the chord of the ellipse $x^2 + \frac{y^2}{4} = 1$ formed on the line $y = x + 1$ is

The area of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac{x^2}{9} + \frac{y^2}{5} = 1$ is .............. $sq. \text{ units}$.

If $B$ is the end point of the minor axis of the ellipse $b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}$ $(a > b)$ and $S$ and $S^{\prime}$ are the foci of the ellipse such that $\Delta SBS^{\prime}$ is an equilateral triangle,then the eccentricity $e$ is:

If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2+4y^2=36$ is $r$,then $12r^2$ is equal to

For $0 < \theta < \frac{\pi}{2}$,four tangents are drawn at the four points $(\pm 3 \cos \theta, \pm 2 \sin \theta)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. If $A(\theta)$ denotes the area of the quadrilateral formed by these four tangents,the minimum value of $A(\theta)$ is