Let $A_1$, $A_2$, $A_3$, ......... $A_8$ 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 $i = 1,2.....8$. If $P$ varies over the circle, then the maximum value of the product $PA_1 \cdot PA_2 .... PA_8$, is

Let A₁, A₂, A₃, ......... A₈ 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 PAᵢ denote the distance between the points P and Aᵢ for i = 1,2.....8. If P varies over the circle, then the maximum value of the product PA₁ · PA₂ .... PA₈, is $\text{Regular octagon } A_1, A_2, \ldots, A_8 \text{ on a circle of radius } R = 2$ $PA_i = \text{distance from } P \text{ (on the circle) to } A_i$ $\text{To find: } \max_{P} \left( PA_1 \cdot PA_2 \cdots PA_8 \right)$ $1.\ \text{put the circle as } |z| = 2 \text{ in the Argand plane}$ $2.\ \text{write the octagon's vertices as } 2\omega^{k-1},\ \omega = e^{i\pi/4}$ $3.\ \text{form } \textstyle\prod PA_k \text{ using } z^{8}-1 = \prod (z-\omega^{j})$ $\text{Circle: } |z| = 2 \quad (\text{centre } O,\ R = 2)$ $P = 2e^{i\theta} \quad (\text{on } |z| = 2)$ $A_k = 2\,\omega^{k-1}, \quad \omega = e^{i\pi/4}, \quad k = 1, \ldots, 8$ $\omega^{8} = 1$ $PA_k = |P - A_k| = 2\left|e^{i\theta} - \omega^{k-1}\right|$ $\prod_{k=1}^{8} PA_k = \prod_{k=1}^{8} 2\left|e^{i\theta} - \omega^{k-1}\right|$ $= 2^{8}\, \prod_{j=0}^{7}\left|e^{i\theta} - \omega^{j}\right| = 256 \left|\prod_{j=0}^{7}\left(e^{i\theta} - \omega^{j}\right)\right|$ $z^{8} - 1 = \prod_{j=0}^{7}\left(z - \omega^{j}\right)$ $\prod_{j=0}^{7}\left(e^{i\theta} - \omega^{j}\right) = e^{8i\theta} - 1$ $\left|e^{8i\theta} - 1\right| = 2\left|\sin 4\theta\right|$ $\prod_{k=1}^{8} PA_k = 256 \cdot 2\left|\sin 4\theta\right| = 512\left|\sin 4\theta\right|$ $\left|\sin 4\theta\right| \le 1 \Rightarrow \max \prod PA_k = 512$ $4\theta = \dfrac{\pi}{2} \Rightarrow \theta = \dfrac{\pi}{8}; \quad \prod PA_k = 512$ $\boxed{\text{Answer: } 512}$