Question

Let $z$ be a complex number satisfying $\overline{z}=\frac{16}{z-i}-i$. Then which of the following is/are true?

• A. Centre of locus of $z$ is $(0,1)$
• B. $|z-i|=16$
• C. Maximum value of $|z|$ is $\sqrt{15}$
• D. Maximum value of $|z|$ is $5$

Answer 1

Answer: (A), (D)

The correct options are
A Centre of locus of $z$ is $(0,1)$
D Maximum value of $|z|$ is $5$

Let $z=x+iy$
Given, $\overline{z}=\frac{16}{z-i}-i$
$\Rightarrow \overline{z}+i=\frac{16}{z-i}\Rightarrow (\overline{z}-\overline{i})(z-i)=16\Rightarrow |z-i{|}^2=16\Rightarrow |z-i|=4\Rightarrow |x+i(y-1)|=4\Rightarrow x^2+(y-1{)}^2=16$
The locus is circle whose centre is $(0,1)$ and radius is $4$
When $x=0,$ $y=-3,5$
When $y=0,$ $x=\pm \sqrt{15}$
The circle intersects $x-$axis at $(\sqrt{15},0)$ and $(-\sqrt{15},0)$
Also intersects $y-$axis at $(0,-3)$ and $(0,5)$
So, the maximum value of $|z|=5$