• # question_answer The radius of an arc given by locus of z if arc $\left( \frac{z-4i}{z-3} \right)=\frac{\pi }{3}$ is A) $5\sqrt{3}$                   B) $3\sqrt{5}$ C) $\frac{5}{\sqrt{3}}$      D) $\frac{5}{2}$

[c] We have, $\arg \left( \frac{z-4i}{z-3} \right)=\frac{\pi }{3}$ In $\Delta OAB,$ $\cos \frac{2\pi }{3}=\frac{O{{A}^{2}}+O{{B}^{2}}-A{{B}^{2}}}{2OA\cdot OB}$ $\Rightarrow$   $\frac{-1}{2}=\frac{2{{r}^{2}}-25}{2{{r}^{2}}}$$\Rightarrow$$3{{r}^{2}}=25$ $r=\frac{5}{\sqrt{3}}$