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

$A$ tangent is drawn to the ellipse $\frac{x^2}{32} + \frac{y^2}{8} = 1$ from the point $A(8, 0)$ to touch the ellipse at point $P$. If the normal at $P$ meets the major axis of the ellipse at point $B$,then the length $BC$ is equal to (where $C$ is the center of the ellipse) - ............ $units$

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

Find the locus of the midpoint of the portion of any tangent to the ellipse $x^{2} + 2y^{2} = 2$ intercepted between the axes.

The eccentricity of the ellipse $(x-3)^2 + (y-4)^2 = \frac{y^2}{9} + 16$ is -

The area of the triangle formed by the tangent and normal to the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at the point $P\left(\frac{a}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$ with the $x$-axis is