The area of the region bounded by the curve $x = 4 \cos \theta, y = 3 \sin \theta$ is . . . . . . sq. units. (in $\pi$)

An ellipse has $OB$ as semi-minor axis,$S$ and $S^{\prime}$ are foci and angle $\angle SBS^{\prime}$ is a right angle. Then the eccentricity of the ellipse is

The normal drawn at the point $\left(\sqrt{9} \cos \frac{\pi}{4}, \sqrt{7} \sin \frac{\pi}{4}\right)$ to the ellipse $\frac{x^2}{9} + \frac{y^2}{7} = 1$ intersects its major axis at the point

The minimum length of the intercept between the coordinate axes made by a tangent of the ellipse $\frac{x^2}{64}+\frac{y^2}{4}=1$ is

The equation of the ellipse whose foci are $(\pm 5, 0)$ and one of its directrices is $5x = 36$ is:

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