Question

If $p\left(\frac{\pi }{6}\right)$ is a point on the ellipse $\frac{x^2}{36}+\frac{y^2}{9}=1,S,S^{\prime }$ are foci of ellipse then $\left|Sp-S^{\prime }p\right|=$

• A. $6\sqrt{3}$
• B. $9$
• C. $9\sqrt{3}$
• D. $27$

Answer 1

The correct option is B $9$

Given: ellipse$=\frac{x^2}{36}+\frac{y^2}{9}=1$, $(a=6,b=3)$

Point $P:p\left(\frac{\pi }{6}\right)$

$\Rightarrow P:(6\cos \frac{\pi }{6},3\sin \frac{\pi }{6})$

$\Rightarrow P:(3\sqrt{3},\frac{3}{2})$

Foci $S$ and $S^{\prime }$

$e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{\frac{27}{36}}=\frac{\sqrt{3}}{2}$

$\Rightarrow S:(3\sqrt{3},0),S^{\prime }:(-3\sqrt{3},0)$

$SP=\sqrt{0^2+(-\frac{3}{2}{)}^2}=\frac{3}{2}$

$S^{\prime }P=\sqrt{{\left(6\sqrt{3}\right)}^2+(\frac{3}{2}{)}^2}=\frac{21}{2}$

$|SP-S^{\prime }P|=|\frac{3}{2}-\frac{21}{2}|=|-9|=9$