Question

If principal argument of $z_0$ satisfying $|z-3| \leq \sqrt{2}$ and $\arg (z-5 i)=\frac{-\pi}{4}$ simultaneously is $\theta$ then identify the correct statement(s)?

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

Answer 1

Answer: (B), (D)

$\arg ( z -5 i )=\frac{-\pi}{4}$ is a ray emanating from $(0,5)$ and making an angle $\frac{\pi}{4}$ from the $x$-axis in clockwise direction.
$\Rightarrow \arg (x+i(y-5))=\frac{-\pi}{4} \Rightarrow \tan ^{-1}\left(\frac{y-5}{x}\right)=\frac{-\pi}{4} \Rightarrow y-5=-x \Rightarrow x+y=5$
Also, $| z -3| \leq \sqrt{2} \Rightarrow|( x -3)+ iy | \leq \sqrt{2} \Rightarrow( x -3)^2+ y ^2 \leq 2$
$\Rightarrow x + y =5$ touches the circle. Let $z _0$ be the complex number corresponding to point of contact.
Let $ z _0= x _0+ iy _0$

$x _0 =3+\sqrt{2} \cos \frac{\pi}{4}=4 $
$y _0= \sqrt{2} \sin \frac{\pi}{4}=1 $
$\therefore z _0 =4+ i$
$\text { Now, } \tan \theta =\frac{1}{4}$
$\Rightarrow \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^2 \theta}=\frac{2 \cdot \frac{1}{4}}{1-\frac{1}{16}}=\frac{8}{15}$
Also, $\left| z _0\right|=\sqrt{17}$
and $\left|z_0-5 i\right|=|4+i-5 i|=4|1-i|=4 \sqrt{2}$.