Let A1, A2, A3........... An be the vertices a regular polygon of n-sides whose centre is origin O. Let the complex number representing verticesA1, A2, A3........... An be z1, z2, z3........... zn respectively. Such thatOA1 = OA2 = ------ = OAn = 1.Then which of the following is/are correct?

A 1 A 2 ⋅ A 1 A 3 … … … . A 1 A n = n , If 1 A 1 A 2 = 1 A 1 A 3 + 1 A 1 A 4 then n = 7 , The length of A 1 A 10 = sin 9 π n

Let A1, A2, A3........... An be the vertices a regular polygon of n-sides whose centre is origin O. Let the complex number representing verticesA1, A2, A3........... An be z1, z2, z3........... zn respectively. Such thatOA1 = OA2 = ------ = OAn = 1.Then which of the following is/are correct?

A) A 1 A r 2 = z 1 − z r 2 = 2 − 2 cos ⁡ 2 ( r − 1 ) π n = 2 n B) A 1 A r 2 = z 1 2 2 − 2 cos 2 ⁡ ( r − 1 ) π n n = 7 C) A 1 A r = z 1 1 − e 2 ( r − 1 ) π / n z n − 1 z − 1 = z − e i 2 π / n z − e i 4 π / n D) From a triangle OA 1 A 10 , ∠ OA 1 A 10 = π 2 − 9 π n By sine rule OA 1 cos ⁡ 9 π n = A 1 A 10 sin ⁡ 18 π n ⇒ A 1 A 10 2 sin ⁡ 9 π n

A 1 A 2 ⋅ A 1 A 3 … … … . A 1 A n = n

If 1 A 1 A 2 = 1 A 1 A 3 + 1 A 1 A 4 then n = 7

The length of A 1 A 10 = sin 9 π n

A 1 A 2 2 + A 1 A 3 2 + … … … . . A 1 A n 2 = 2 n

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Let A₁, A₂, A₃........... A_n be the vertices a regular polygon of n-sides whose centre is origin O. Let the complex number representing vertices
A₁, A₂, A₃........... A_n be z₁, z₂, z₃........... z_nrespectively. Such that
OA₁ = OA₂ = ------ = OA_n = 1.Then which of the following is/are correct?

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$|A_{1} A_{2}| \cdot |A_{1} A_{3}| \dots \dots \dots . |A_{1} A_{n}| = n$

If $\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}}$ then n = 7

The length of $A_{1} A_{10} = \sin \frac{9 π}{n}$

${|A_{1} A_{2}|}^{2} + {|A_{1} A_{3}|}^{2} + \dots \dots \dots . . {|A_{1} A_{n}|}^{2} = 2 n$

answer is A, B, C.

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Detailed Solution

$\begin{array}{l} A) {|A_{1} A_{r}|}^{2} = {|z_{1} - z_{r}|}^{2} \\ = 2 - 2 \cos \frac{2 (r - 1) π}{n} = 2 n \end{array} B) {|A_{1} A_{r}|}^{2} = {|z_{1}|}^{2} [2 - 2 \cos^{2} \frac{(r - 1) π}{n}]$
n = 7
$\begin{array}{l} C) |A_{1} A_{r}| = |z_{1}| |1 - e^{2 (r - 1) \sqrt{π / n}}| \\ \frac{z^{n} - 1}{z - 1} = (z - e^{i 2 π / n}) (z - e^{i 4 π / n}) \end{array}$
D) From a triangle ${OA}_{1} A_{10}, ∠ {OA}_{1} A_{10} = \frac{π}{2} - \frac{9 π}{n}$
By sine rule $\frac{{OA}_{1}}{\cos \frac{9 π}{n}} = \frac{A_{1} A_{10}}{\sin \frac{18 π}{n}}$
$\Rightarrow A_{1} A_{10} 2 \sin \frac{9 π}{n}$