JEE Advanced 2023 Paper 2, Mathematics Section 3 Q4: De Moivre's Theorem And Its Applications

JEE Advanced2023Paper 2Mathematics Section 3

Let $A_{1}, A_{2}, A_{3}, \dots, A_{8}$ 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 $i = 1, 2, \dots, 8$ . If $P$ varies over the circle, then the maximum value of the product $P A_{1} \cdot P A_{2} \dots P A_{8}$ , is

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Solution

Place the circle at the origin with radius $2$ . Let the octagon vertices be $A_{k} = 2 ω^{k}$ , where $ω = e^{iπ /4}$ is a primitive eighth root of unity. Then ${A_{k}}$ are the roots of $z^{8} = 2^{8}$ , so

$z^{8} - 2^{8} = k = 0 \prod 7 (z - 2 ω^{k}) .$

Pick a point $P = 2 e^{i θ}$ on the same circle. Substituting $z = 2 e^{i θ}$ gives

$2^{8} (e^{i 8 θ} - 1) = k = 0 \prod 7 (2 e^{i θ} - 2 ω^{k}) .$

Taking moduli, $P A_{1} \cdot P A_{2} \dots P A_{8} = 2^{8} ∣ e^{i 8 θ} - 1∣ = 2^{8} \cdot 2∣ sin 4 θ ∣ = 512∣ sin 4 θ ∣.$

The maximum value is $512$ , attained when $∣ sin 4 θ ∣ = 1.$

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JEE Advanced 2023 Paper 2, Mathematics Section 3 Q4: De Moivre's Theorem And Its Applications

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