Question

If $P\left(\frac{\pi }{4}\right)$ is any point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $S,S^{\prime }$ are its foci, then find
i) Perimeter of $\Delta SP{S}^{\prime }$
ii) $SP.S^{\prime }P$

Answer 1

$a=5,b=3⟹e=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{\frac{25-9}{25}}=\frac{4}{5}$

Foci of the ellipse is given by $(\pm ae,0)=(\pm 4,0)$

Parametric points if Ellipse are $x=a\sin \theta ,y=b\cos \theta$

$⟹P\left(\frac{\pi }{4}\right)=(\frac{5}{\sqrt{2}},\frac{3}{\sqrt{2}})$

$SP=\sqrt{{(4-\frac{5}{\sqrt{2}})}^2+{(0-\frac{3}{\sqrt{2}})}^2}\approx 2.16$

$S^{\prime }P=\sqrt{{(-4-\frac{5}{\sqrt{2}})}^2+{(0-\frac{3}{\sqrt{2}})}^2}\approx 7.86$

$S^{\prime }S=2a=10$

so perimeter $\approx 20$

$SP.S^{\prime }P=\left(2.16\right)\left(7.86\right)=16.977$