Question

If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if $PS{P}^{\prime }$ is a focal chord with $SP=8$, then $S{S}^{\prime }$ is equal to

• A. $4+S^{\prime }P$
• B. $S^{\prime }P-1$
• C. $4+SP$
• D. $SP-1$

Answer 1

Answer: (A)

$4+S^{\prime }P$

Given equation of ellipse is $\frac{x^2}{25}+\frac{y^2}{16}=1$

For the given ellipse $a=5,b=4$

Also given, $SP=8$

We know $e^2=1-\frac{b^2}{a^2}$

$\Rightarrow e^2=1-\frac{16}{25}\Rightarrow e^2=\frac{9}{25}$

$\Rightarrow e=\frac{3}{5}$

Also $S{S}^{\prime }=2ae$

$\Rightarrow S{S}^{\prime }=2\times 5\times \frac{3}{5}=6$

And $SP+S^{\prime }P=2a$

$\Rightarrow 8+S^{\prime }P=2\left(5\right)\Rightarrow S^{\prime }P=2\Rightarrow S{S}^{\prime }=6\Rightarrow S{S}^{\prime }=4+2\Rightarrow S{S}^{\prime }=4+S^{\prime }P$

So, option A is correct.