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) \[\frac{5}{2}\]

Correct Answer: C

Solution :

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