If S and S^' are the foci of the ellipse x^218+y^29=1 and P be a point on the ellipse, the…

Question

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

Accepted Answer

$$\begin{aligned}
& a=3 \sqrt{2}, b=3 \
& \Rightarrow \quad e=\frac{1}{\sqrt{2}}
\end{aligned}$$

$$\begin{aligned}
& P S \cdot P S^{\prime}=2 a=6 \sqrt{2} \
& \frac{P S+P S^{\prime}}{2} \geq \sqrt{P S \cdot P S^{\prime}} \
& \Rightarrow\left(P S \times P S^{\prime}\right) \max =18
\end{aligned}$$

Minima happens when $P$ lies on major axis

$$\begin{aligned}
& \Rightarrow \quad P=(3 \sqrt{2}, 0) \
& P S=(3 \sqrt{2}-3) \cdot P S^{\prime}=(3 \sqrt{2}+3) \
& \left(P S \cdot P S^{\prime}\right) \min =9 \
& \left(P S \cdot P S^{\prime}\right) \min +\left(P S \cdot P S^{\prime}\right) \max =27
\end{aligned}$$

Option (1)

Solutions · 1