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 P A 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 P A 1 · P A 2 … · P A 8 , is
A. A
Given, A 1 , A 2 , A 3 , … , A 8 vertices of a regular octagon lying on a circle of radius 2 . Now using the concept of n t h root of unity, Let any point P be, Z = ( 2 ) ( 1 ) 1 / 8 ⇒ Z 8 = 2 8 × 1 ⇒ Z 8 − 2 8 = 0 ⇒ Z = 2 , 2 α , 2 α 2 , 2 α 3 , … , 2 α 7 ; here  α = e i 2 π 8 ⇒ Z 8 − 2 8 = ( Z − 2 ) ( Z − 2 α ) Z − 2 α 2 Z − 2 α 3 … Z − 2 α 7 ⇒ Z 8 − 2 8 = | Z − 2 | | Z − 2 α | … . Z − 2 α 7  But  Z 8 + − 2 8 ≤ | Z | 8 + 2 8 ⇒ | Z − 2 | | Z − 2 α | … Z − 2 α 7 ≤ | Z | 8 + 2 8 ≤ 2 8 + 2 8         ≤ 2 9                         ⇒ Max ⁡ P A 1 ⋅ P A 2 … . P A 8 = 2 9 = 512
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