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 PAi 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 8th root of 28 & let P be 2eiα.
Now, z8 – 28 = (z – A1) (z – A2) ……… (z – A8)
So, Put z = 2eiα
⇒ 28ei8α – 28 = (2eiα – A1) (2eiα – A2) (2eiα – A3) ….. (2eiα – A8)
⇒ 28 |ei8α – 1| = |(2eiα – A1) (2eiα – A2) (2eiα – A3) ….. (2eiα – A8)|
⇒ 28 |ei4α – e–i4α | = |(2eiα – A1) (2eiα – A2) (2eiα – A3) ….. (2eiα – A8)|
⇒ 29 |sin4α| = |(2eiα – A1) (2eiα – A2) (2eiα – A3) ….. (2eiα – A8)|
So, from this we get that :
Maximum value of the PA1.PA2 …… PA8 = 29
Therefore, the correct answer is 29 = 512.