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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 ${PA}_{1} \cdot {PA}_{2} \dots . {PA}_{8}$ , is

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answer is 512.

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Detailed Solution

$z^{8} - 2^{8} = (z - 2) (z - 2 α) (z - 2 α^{2}) . ... (z - 2 α^{7})$
$|z - 2| |z - 2 α| . ......... |z - 2 α^{7}| = |z^{8} - 2^{8}|$
$≤ {|z|}^{8} + 2^{8}$
$\leq 2^{8} (2)$
$\leq 2^{9}$

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