Let $P$ be a variable point on the ellipse $b^2{x}^2+a^2{y}^2=a^2{b}^2$ with foci $F_1$ and $F_2,$ then the maximum area of $\Delta P{F}_1{F}_2$ is

If tangents are drawn from a point on $x^2-4y^2=4$ to $x^2+4y^2=4$ touching it at $Q$ and $R,$ then mid-point of $QR$ lies on

A normal of slope $1$ of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major and minor axes at $P$ and $Q$ respectively. If $C$ be the centre of ellipse, then the area of $\Delta CPQ$ is

If the eccentric angles of the extremities of a chord of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $\alpha$ and $\beta$ and if the chord cuts the major axis at a distance $c$ from centre, then $\tan \left(\frac{\alpha }{2}\right)\tan \left(\frac{\beta }{2}\right)$ is equal to