Question

If $|z-25i|\le 15$ then $|$ maximum amp $\left(z\right)-$ minimum amp $\left(z\right)|$ is equal to

• A. ${\sin }^{-1}\left(\frac{3}{5}\right)-{\cos }^{-1}\left(\frac{3}{5}\right)$
• B. $\frac{\pi }{2}+{\cos }^{-1}\left(\frac{3}{5}\right)$
• C. $\pi -2{\cos }^{-1}\left(\frac{3}{5}\right)$
• D. ${\cos }^{-1}\left(\frac{3}{5}\right)$

Answer 1

The correct option is C $\pi -2{\cos }^{-1}\left(\frac{3}{5}\right)$

We have $maximum.amp\left(z\right)$=$=amp\left(z_2\right)$

$mini.amp\left(z\right)=amp\left(z_1\right)$

Now $amp\left(z_1\right)={\theta }_1={\cos }^{-1}\frac{15}{25}={\cos }^{-1}\frac{3}{5}$

and $amp\left(z_2\right)=\frac{\pi }{2}+{\theta }_2=\frac{\pi }{2}+{\sin }^{-1}\frac{15}{25}=\frac{\pi }{2}+{\sin }^{-1}\frac{3}{5}$

So $|max.amp(z)-mi.amp(z\left)\right|=\left|\frac{\pi }{2}+{\sin }^{-1}\frac{3}{5}-{\cos }^{-1}\frac{3}{5}\right|$

$=\left|\frac{\pi }{2}+\frac{\pi }{2}-2{\cos }^{-1}\frac{3}{5}\right|=\pi -2{\cos }^{-1}\frac{3}{5}$