Complex Numbers And Quadratic Equations question 143
Question: If $ |z-25i|\le 15 $ , then $ |\max .amp(z)-\min .amp(z)|= $
Options:
A) $ {{\cos }^{-1}}( \frac{3}{5} ) $
B) $ \pi -2{{\cos }^{-1}}( \frac{3}{5} ) $
C) $ \frac{\pi }{2}+{{\cos }^{-1}}( \frac{3}{5} ) $
D) $ {{\sin }^{-1}}( \frac{3}{5} )-{{\cos }^{-1}}( \frac{3}{5} ) $
Show Answer
Answer:
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}}( \frac{15}{25} )={{\cos }^{-1}}( \frac{3}{5} ) $ $ amp(z_2)=\frac{\pi }{2}+{\theta_2}=\frac{\pi }{2}+{{\sin }^{-1}}( \frac{15}{25} )=\frac{\pi }{2}+{{\sin }^{-1}}( \frac{3}{5} ) $ \ $ |\max amp(z)-\min amp(z)| $ $ =| \frac{\pi }{2}+{{\sin }^{-1}}\frac{3}{5}-{{\cos }^{-1}}\frac{3}{5} | $ $ =| \frac{\pi }{2}+\frac{\pi }{2}-{{\cos }^{-1}}\frac{3}{5}-{{\cos }^{-1}}\frac{3}{5} |=\pi -2{{\cos }^{-1}}\frac{3}{5} $
BETA
