Question

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

Answer 1

The given ellipse is $9x^{2}+36y^{2}=324$
$\Rightarrow \frac{9 x^{2}}{324}+\frac{36 y^{2}}{324}=1$
$\Rightarrow \frac{x^{2}}{36}+\frac{y^{2}}{9}=1$
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 $S^{'}$ and $P$ is any point on the ellipse, then by the definition of the ellipse, we have
$SP+S'P=2a=2\times 6=12$