If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the ellipse, then $\min \left(S P . S^{\prime} \mathrm{P}\right)+$ $\max \left(\mathrm{SP} . \mathrm{S}^{\prime} \mathrm{P}\right)$ is equal to :

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Aicharya · Accepted Answer

$\mathrm{PS}+\mathrm{PS}^{\prime}=2 	imes 3 \sqrt{2}$ $\mathrm{b}^2=\mathrm{a}^2\left(1-\mathrm{e}^2ight) \Rightarrow 9=18\left(1-\mathrm{e}^2ight)$ $\Rightarrow \mathrm{e}=\frac{1}{\sqrt{2}}$ Directrix $\mathrm{x}=\frac{\mathrm{a}}{\mathrm{e}}=\frac{3 \sqrt{2}}{\frac{1}{\sqrt{2}}}=6$ $P S \cdot P S '=\left|\frac{1}{\sqrt{2}}(3 \sqrt{2} \cos 	heta-6) \frac{1}{\sqrt{2}}(3 \sqrt{2} \cos 	heta+6)ight|$ $=\frac{1}{2}\left|18 \cos ^2 	heta-36ight|$ $\left(\mathrm{PS} \cdot \mathrm{PS}^{\prime}ight)_{\max }=18 ;(\mathrm{PS} \cdot \mathrm{PS})_{\min }=9$ sum $=27$

If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the…

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