If a tangent with slope $-4/3$ to the ellipse $\frac{x^2}{18} + \frac{y^2}{32} = 1$ intersects the major axis and minor axis at $A$ and $B$ respectively,then the area of $\Delta OAB$ is .......... square units.

Number of points on the ellipse $\frac{x^2}{50} + \frac{y^2}{20} = 1$ from which a pair of perpendicular tangents are drawn to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is:

Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$,pass through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and have eccentricity $e = \frac{1}{\sqrt{3}}$. If a circle,centered at the focus $F(\alpha, 0), \alpha > 0$ of $E$ and having radius $r = \frac{2}{\sqrt{3}}$,intersects $E$ at two points $P$ and $Q$,then $PQ^{2}$ is equal to:

The equations of the tangents to the ellipse $9x^2 + 16y^2 = 144$ which pass through the point $(2, 3)$ are

Let $A_1$ be the area of the given ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let $A_2$ be the area of the region bounded by the curve which is the locus of the midpoint of the line segment joining the focus of the ellipse and a point $P$ on the given ellipse. Then $A_1 : A_2$ is equal to:

If the length of the minor axis of an ellipse is equal to half of the distance between the foci,then the eccentricity of the ellipse is: