JEE Advanced 2023 · Paper 2 · Mathematics Q11

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MathematicsInteger+4Question 11 of 51

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

Your answer

Answer and worked solution

Official answer: 512

Place the circle at the origin, with $A_k=2\omega^k$ where $\omega=e^{2\pi i/8}$ . Let $P=2e^{i\theta}$ .

Product of distances. Since $\prod_k(z-2\omega^k)=z^8-2^8$ ,
$PA_1\cdots PA_8=|P^8-2^8|=2^8\left|e^{8i\theta}-1\right|.$

Maximum. $|e^{8i\theta}-1|\le2$ , with equality when $e^{8i\theta}=-1$ . This is when $P$ is midway between two vertices.

$\max=2\cdot2^8=512.$

Topics:Roots of unity Geometry in the complex plane

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JEE Advanced 2023 · Paper 2, Mathematics question 11