Question

Let $P_1 P_2 P_3..... P_{10}$ be vertices of a regular polygon lying on argand plane of 10 sides whose centre is at the origin O. Let the complex numbers representing vertices $P_1, P_2, P_3 .....$ and $P_{10}$ be $w_1, w_2, w_3.......$ and $w_{10}$ respectively. Let $OP_1 = OP_2 = OP_3 = ...... = OP_{10} = 1$

Then value of the product $|P_1 P_2| \times |P_1 P_3| \times |P_1 P_4| \times ..... \times |P_1 P_{10}|$ is equal to

Answer 1

10

Answer 2

The vertices $w_k$ are roots of $z^{10}=1$. The product is $\prod_{k=2}^{10} |w_1-w_k|$. This equals $|\prod_{k=2}^{10} (w_1-w_k)|$. Using $w_k=w_1\zeta^{k-1}$, the product becomes $|w_1|^9 \prod_{j=1}^9 |1-\zeta^j|$. Since $|w_1|=1$, it is $\prod_{j=1}^9 |1-\zeta^j|$. The roots $\zeta^j$ for $j=1,\dots,9$ are roots of $\frac{z^{10}-1}{z-1}=z^9+\dots+1$. Evaluating at $z=1$ gives $10 = \prod_{j=1}^9 (1-\zeta^j)$. Taking modulus yields 10.