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 value of $SP\cdot S^{\prime }P$ is:

• A. $9\le f\left(\theta \right)\le 16$
• B. $9\le f\left(\theta \right)\le 25$
• C. $16\le f\left(\theta \right)\le 25$
• D. $1\le f\left(\theta \right)\le 16$

Answer 1

The correct option is C $16\le f\left(\theta \right)\le 25$

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

where, $e$ is the eccentricity of the ellipse

Now,

$(SP-S^{\prime }P)(SP+S^{\prime }P)=25-25e^2{\cos }^2\theta$
$=25{\sin }^2\theta +16{\cos }^2\theta$
$\Rightarrow 16\le f\left(\theta \right)\le 25$

Hence, option C.