If $\alpha$ and $\beta$ are the eccentric angles of the endpoints of a focal chord of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2} = ....$

The equation $\frac{x^2}{10-a} + \frac{y^2}{4-a} = 1$ represents an ellipse when:

$A$ 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 line $lx + my + n = 0$ will cut the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at points whose eccentric angles differ by $\pi/2$ if:

Find the area of the quadrilateral formed by the tangents at the endpoints of the latus rectum of the ellipse $x^2 + 2y^2 = 2$.

If $x+\sqrt{3} y=3$ is the tangent to the ellipse $2 x^2+3 y^2=k$ at a point $P$,then the equation of the normal to this ellipse at $P$ is