JEE Advance - Mathematics (2023 - Paper 2 Online - No. 11)
Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius 2 . Let $P$ be a point on the circle and let $P A_i$ denote the distance between the points $P$ and $A_i$ for $i=1,2, \ldots, 8$. If $P$ varies over the circle, then the maximum value of the product $P A_1 \times P A_2 \times \cdots \cdots \times P A_8$, is :
Answer
512
Explanation
$A_1, A_2, A_3, \ldots, A_8$ vertices of a regular octagon lying on a circle of radius 2 .
Let say, $Z=(2)(1)^{1 / 8}$
$$ \begin{aligned} & \Rightarrow Z^8=2^8 \times 1 \\\\ & \Rightarrow Z^8-2^8=0 \end{aligned} $$
$\begin{aligned} & \Rightarrow Z=2,2 \alpha, 2 \alpha^2, 2 \alpha^3, \ldots, 2 \alpha^7 ; \alpha=e^{i \frac{2 \pi}{8}} \\\\ & \Rightarrow Z^8-2^8=(Z-2)(Z-2 \alpha)\left(Z-2 \alpha^2\right)\left(Z-2 \alpha^3\right) \ldots\left(Z-2 \alpha^7\right) \\\\ & \Rightarrow\left|Z^8-2^8\right|=|Z-2||Z-2 \alpha| \ldots .\left|Z-2 \alpha^7\right| \\\\ & \text { But }\left|Z^8+\left(-2^8\right)\right| \leq|Z|^8+2^8\end{aligned}$
$\begin{aligned} \Rightarrow|Z-2||Z-2 \alpha| \ldots\left|Z-2 \alpha^7\right| & \leq|Z|^8+2^8 \\\\ & \leq 2^8+2^8 \\\\ & \leq 2^9\end{aligned}$
$\Rightarrow \operatorname{Max}\left(P A_1 \cdot P A_2 \ldots P A_8\right)=2^9$
Let say, $Z=(2)(1)^{1 / 8}$
$$ \begin{aligned} & \Rightarrow Z^8=2^8 \times 1 \\\\ & \Rightarrow Z^8-2^8=0 \end{aligned} $$
$\begin{aligned} & \Rightarrow Z=2,2 \alpha, 2 \alpha^2, 2 \alpha^3, \ldots, 2 \alpha^7 ; \alpha=e^{i \frac{2 \pi}{8}} \\\\ & \Rightarrow Z^8-2^8=(Z-2)(Z-2 \alpha)\left(Z-2 \alpha^2\right)\left(Z-2 \alpha^3\right) \ldots\left(Z-2 \alpha^7\right) \\\\ & \Rightarrow\left|Z^8-2^8\right|=|Z-2||Z-2 \alpha| \ldots .\left|Z-2 \alpha^7\right| \\\\ & \text { But }\left|Z^8+\left(-2^8\right)\right| \leq|Z|^8+2^8\end{aligned}$
$\begin{aligned} \Rightarrow|Z-2||Z-2 \alpha| \ldots\left|Z-2 \alpha^7\right| & \leq|Z|^8+2^8 \\\\ & \leq 2^8+2^8 \\\\ & \leq 2^9\end{aligned}$
$\Rightarrow \operatorname{Max}\left(P A_1 \cdot P A_2 \ldots P A_8\right)=2^9$
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