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.

If the focal distance of an endpoint of the minor axis of an ellipse (taking its axes as the $x$ and $y$ axes respectively) is $k$ and the distance between its foci is $2h$,then its equation is:

$A$ tangent is drawn to the ellipse $\frac{x^2}{27} + y^2 = 1$ at the point $(3\sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in (0, \pi/2)$. The value of $\theta$ for which the sum of the intercepts on the axes made by this tangent is minimum,is:

Let $A$ be a vertex of the ellipse $S \equiv \frac{x^2}{4}+\frac{y^2}{9}-1=0$ and $F$ be a focus of the ellipse $S^{\prime} \equiv \frac{x^2}{9}+\frac{y^2}{4}-1=0$. Let $P$ be a point on the major axis of the ellipse $S^{\prime}=0$,which divides $\overline{OF}$ in the ratio $2:1$ ($O$ is the origin). If the length of the chord of the ellipse $S=0$ through $A$ and $P$ is $\frac{3\sqrt{101}}{k}$,then $k=$

If the line $3x + 4y + k = 0$ touches the ellipse $9x^2 + 16y^2 = 144$, then the value of $k$ is:

Given the base of a triangle and the sum of its other two sides,the locus of the center of its incircle is: