Let $λ, z_{0}$ be two complex numbers. $A (z_{1}), B (z_{2}), C (z_{3})$ be the vertices of a triangle such that $z_{1} = z_{0} + λ, z_{2} = z_{0} + λ e^{i π / 4}, a n d$ $z_{3} = z_{0} + λ e^{i 7 π / 11}$ and $∠ A B C = k π$ , then the value of $k$ is

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Detailed Solution

$| z_{1} - z_{0} | = | z_{2} - z_{0} | = | z_{3} - z_{0} | = | λ | \frac{z_{1} - z_{0}}{z_{2} - z_{0}} = \frac{e^{i 7 π / 11}}{e^{i π / 4}} = e^{i 17 / 14} \Rightarrow ∠ B S C = 17 \frac{π}{44} \Rightarrow ∠ B A C = 14 \frac{π}{88} S i m i l a r l y, \frac{z_{2} - z_{0}}{z_{1} - z_{0}} = e^{i π / 4} \Rightarrow ∠ A C B = \frac{π}{8} ∴ ∠ A B C = π - \frac{π}{8} - \frac{17 π}{88} = \frac{15 π}{22}$