JEE Advanced 2023 Paper 2Question 1
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 $PA_i$ 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 $PA_1\cdot PA_2\cdot\ldots\cdot PA_8$, is
Complex Numbers
Answer$512$
Solution
With the vertices as roots of $z^8=2^8$, $\prod PA_i=|z^8-256|$ for $|z|=2$, which is maximal ($=512$) when $z^8=-256$.