Let A _ 1 , A _ 2 , A _ 3 , , A _ 8 be the vertices of a regular...

Given,

${A}_{1}, {A}_{2}, {A}_{3}, \ldots , {A}_{8}$ 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, $Z = ( 2 ) ( 1{)}^{1 / 8}$

$⇒ {Z}^{8} = {2}^{8} \times 1$

$⇒ {Z}^{8} - {2}^{8} = 0$

$⇒ Z = 2, 2\alpha, 2{\alpha}^{2}, 2{\alpha}^{3}, \ldots , 2{\alpha}^{7}; \left\{\text{here}\alpha = {e}^{i\dfrac{2\pi}{8}}\right\}$

$⇒ {Z}^{8} - {2}^{8} = ( Z - 2 ) ( Z - 2\alpha ) \left(Z - 2{\alpha}^{2}\right)\left(Z - 2{\alpha}^{3}\right) \ldots \left(Z - 2{\alpha}^{7}\right)$

$⇒ \left|{Z}^{8} - {2}^{8}\right| = | Z - 2 | | Z - 2\alpha | \ldots . \left|Z - 2{\alpha}^{7}\right|$

$\text{But}\left|{Z}^{8} + \left(- {2}^{8}\right)\right| \le | Z{|}^{8} + {2}^{8}$

$$\begin{array}{ll}⇒ | Z - 2 | | Z - 2\alpha | \ldots \left|Z - 2{\alpha}^{7}\right|\le | Z{|}^{8} + {2}^{8} \\ \le {2}^{8} + {2}^{8} \\ \le {2}^{9}\end{array}$$

$⇒ Max \left(P{A}_{1} ⋅ P{A}_{2} \ldots . P{A}_{8}\right) = {2}^{9} = 512$