Question

If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,$ and $P$ is any point on it then range of values of $SP\cdot S^{\prime }P$ is

• A. $[9,16]$
• B. $[9,25]$
• C. $[16,25]$
• D. $[1,10]$

Answer 1

The correct option is C $[16,25]$

Let $P(5\cos \theta ,4\sin \theta )$ be any point on the ellipse whose $e=\frac{3}{5}$
Then
$SP=5+5e\cos \theta$
and $S^{\prime }P=5-5e\cos \theta$

$\Rightarrow SP\cdot S^{\prime }P=25-25e^2{\cos }^2\theta$
So, let $f\left(\theta \right)\Rightarrow SP\cdot S^{\prime }P=25-9{\cos }^2\theta$
$\therefore 16\le f\left(\theta \right)\le 25$