Question

$P$ is a variable point on ellipse $9x^2+16y^2=144$ with foci $S$ and $S^{\prime }.$ If $K$ is the area of triangle $S{S}^{\prime }P,$ then the maximum value of $K^2$ is

Answer 1

Equation of the ellipse is
$9x^2+16y^2=144\Rightarrow \frac{x^2}{16}+\frac{y^2}{9}=1$
$\therefore a=4,b=3$
and $e=\sqrt{1-\frac{9}{16}}=\frac{\sqrt{7}}{4}$
Distance between foci $S{S}^{\prime }=2ae=2\sqrt{7}$
Any point on the ellipse is $P(4\cos \theta ,3\sin \theta )$
$\therefore$ Area of $△S{S}^{\prime }P=\frac{1}{2}\times S{S}^{\prime }\times 3\sin \theta$
$\Rightarrow K=3\sqrt{7}\sin \theta$
$\Rightarrow K^2=63{\sin }^2\theta$
$\therefore K_{max}^2=63$