Question

Let A₁, A₂, A₃, A₄,........, A₈ be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PA_i denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA₁.PA₂..........PA₈, is 

Answer 1

According to the question, Ai are 8^th root of 2⁸ & let P be 2e^iα.
Now, z⁸ – 2⁸ = (z – A₁) (z – A₂) ……… (z – A₈)
So, Put z = 2e^iα
⇒ 2⁸e^i8α – 2⁸ = (2e^iα – A₁) (2e^iα – A₂) (2e^iα – A₃) ….. (2e^iα – A₈)
⇒ 2⁸ |e^i8α – 1| = |(2e^iα – A₁) (2e^iα – A₂) (2e^iα – A₃) ….. (2e^iα – A₈)|
⇒ 2⁸ |e^i4α – e^–i4α | = |(2e^iα – A₁) (2e^iα – A₂) (2e^iα – A₃) ….. (2e^iα – A₈)|
⇒ 2⁹ |sin4α| = |(2e^iα – A₁) (2e^iα – A₂) (2e^iα – A₃) ….. (2e^iα – A₈)|
So, from this we get that :
Maximum value of the PA₁.PA₂ …… PA₈ = 2⁹
Therefore, the correct answer is 2⁹ = 512.