question_answer

If  \[|z-25i|\le 15\],  then \[|\max .amp(z)-\min .amp(z)|=\]

A) \[{{\cos }^{-1}}\left( \frac{3}{5} \right)\]

B) \[\pi -2{{\cos }^{-1}}\left( \frac{3}{5} \right)\]

C) \[\frac{\pi }{2}+{{\cos }^{-1}}\left( \frac{3}{5} \right)\]

D) \[{{\sin }^{-1}}\left( \frac{3}{5} \right)-{{\cos }^{-1}}\left( \frac{3}{5} \right)\]

Correct Answer: B

Solution :

We have max amp(z)=amp\[({{z}_{2}}),\]min amp (z)=amp\[({{z}_{1}})\] Now \[amp({{z}_{1}})={{\theta }_{1}}={{\cos }^{-1}}\left( \frac{15}{25} \right)={{\cos }^{-1}}\left( \frac{3}{5} \right)\] \[amp({{z}_{2}})=\frac{\pi }{2}+{{\theta }_{2}}=\frac{\pi }{2}+{{\sin }^{-1}}\left( \frac{15}{25} \right)=\frac{\pi }{2}+{{\sin }^{-1}}\left( \frac{3}{5} \right)\] \ \[|\max \,\,amp(z)-\min \,\,amp(z)|\] \[=\left| \frac{\pi }{2}+{{\sin }^{-1}}\frac{3}{5}-{{\cos }^{-1}}\frac{3}{5} \right|\] \[=\left| \frac{\pi }{2}+\frac{\pi }{2}-{{\cos }^{-1}}\frac{3}{5}-{{\cos }^{-1}}\frac{3}{5} \right|=\pi -2{{\cos }^{-1}}\frac{3}{5}\]