Question

For a point $P$ on the ellipse $9x^2+36y^2=324$ , with foci $S$ and $S^{{}^{\prime }}$ , find value of $SP+S^{{}^{\prime }}P$ .

Answer 1

Answer: 12

The given ellipse is $9x^2+36y^2=324$
$\Rightarrow \frac{9x^2}{324}+\frac{36y^2}{324}=1$
$\Rightarrow \frac{x^2}{36}+\frac{y^2}{9}=1$
The ellipse is of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b,$
Hence, we have $a=6,b=3$
Also, we know that if the foci are $S$ and $S^{{}^{\prime }}$ and $P$ is any point on the ellipse, then by the definition of the ellipse, we have
$SP+S^{\prime }P=2a=2\times 6=12$