Question

If principal argument of $z_0$ satisfying $|z-3|\le \sqrt{2}$ and $\arg (z-5i)=-\frac{\pi }{4}$ simultaneously is $\theta ,$ then the CORRECT statement(s) is/are

• A. $|z_0|=17$
• B. $\tan 2\theta =\frac{8}{15}$
• C. $\tan \theta =-\frac{1}{4}$
• D. $|z_0-5i|=4\sqrt{2}$

Answer 1

Answer: (B), (D)

$|z_0-5i|=4\sqrt{2}$


Let $z=x+iy$
$\Rightarrow \arg (x+i(y-5\left)\right)=-\frac{\pi }{4}$
$\Rightarrow {\tan }^{-1}\left(\frac{y-5}{x}\right)=-\frac{\pi }{4}$
$\Rightarrow y-5=-x$
$\Rightarrow x+y=5$

Also, $|z-3|\le \sqrt{2}$
$\Rightarrow (x-3{)}^3+y^2\le 2$
Let $z_0$ be point of contact of line and circle.
By solving equations of circle and line, we get
$z_0=4+i$
$\tan \theta =\frac{1}{4}\Rightarrow \tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{8}{15}$
and $|z_0|=\sqrt{17},$ $|z_0-5i|=4\sqrt{2}$