Question

$PSQ$ is a focal chord of the ellipse $4x^2+9y^2=36$ such that $SP=4$. If $S^{\prime }$ is the another focus, write the value of $S^{\prime }Q$

Answer 1

Given that $PSQ$ is the focal chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$

$\Rightarrow a^2=9$ or $a=3$ and $b^2=4$ or $b=2$

We know that semi latus rectum is the harmonic mean of any two segments of focal chord.

$\Rightarrow \frac{2}{\frac{b^2}{a}}=\frac{1}{SP}+\frac{1}{SQ}$

$\Rightarrow \frac{2\times 3}{4}=\frac{1}{4}+\frac{1}{SQ}$

$\Rightarrow \frac{1}{SQ}=\frac{5}{4}$

$\Rightarrow SQ=\frac{4}{5}$

We know that $SQ+S^{\prime }Q=2a$

$\Rightarrow \frac{4}{5}+S^{\prime }Q=2\times 3$

$\Rightarrow S^{\prime }Q=6-\frac{4}{5}=\frac{30-4}{5}=\frac{26}{5}$

$\therefore S^{\prime }Q=\frac{26}{5}$