Question

If $S$ and $S^{\prime }$ are two foci of the ellipse $16x^2+25y^2=400$ and $PSQ$ is a focal chord such that $SP=16,$ then $S^{\prime }Q=$

• A. $\frac{16}{9}$
• B. $\frac{74}{9}$
• C. $\frac{37}{9}$
• D. $\frac{32}{3}$

Answer 1

The correct option is B $\frac{74}{9}$

Equation of the ellipse is
$\frac{x^2}{25}+\frac{y^2}{16}=1.$

We know that if $PSQ$ is a focal chord of ellipse, then length of semi latus rectum is the harmonic mean of $SP$ and $SQ.$
$\Rightarrow \frac{1}{SP}+\frac{1}{SQ}=\frac{2a}{b^2}$
$\Rightarrow \frac{1}{16}+\frac{1}{SQ}=2\times \frac{5}{16}$
$\Rightarrow \frac{1}{SQ}=\frac{9}{16}$
Now $SQ+S^{\prime }Q=2a=10$
$\Rightarrow S^{\prime }Q=10-\frac{16}{9}=\frac{74}{9}$