Let $A_1,A_2,A_3,\dots ,A_8$ 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 $i=1,2,\dots ,8$. If $P$ varies over the circle, then the maximum value of the product $P{A}_1·P{A}_2\dots ·P{A}_8$, is

Given,
$A_1,A_2,A_3,\dots ,A_8$ vertices of a regular octagon lying on a circle of radius $2$.
Now using the concept of $n^{th}$ root of unity,
Let any point $P$ be, $Z=\left(2\right)(1{)}^{1/8}$
\(\Rightarrow Z^8=2^8 \times 1 \)
\( \Rightarrow Z^8-2^8=0 \)
\( \Rightarrow Z=2,2 \alpha, 2 \alpha^2, 2 \alpha^3, \ldots, 2 \alpha^7 ;\left\{\text { here } \alpha=e^{i \frac{2 \pi}{8}}\right\} \)
\( \Rightarrow Z^8-2^8=(Z-2)(Z-2 \alpha)\left(Z-2 \alpha^2\right)\)\(\left(Z-2 \alpha^3\right) \ldots\left(Z-2 \alpha^7\right) \)
\( \Rightarrow\left|Z^8-2^8\right|=|Z-2||Z-2 \alpha| \ldots\left|Z-2 \alpha^7\right| \)
\( \text { But }\left|Z^8+\left(-2^8\right)\right| \leq|Z|^8+2^8 \)
\( \Rightarrow|Z-2||Z-2 \alpha| \ldots\left|Z-2 \alpha^7\right| \leq|Z|^8+2^8 \)
\( \leq 2^8+2^8 \)
\( \leq 2^9 \)
\( \Rightarrow \operatorname{Max}\left(P A_1 \cdot P A_2 \ldots P A_8\right)=2^9=512\)