Question

The complex number $z$ satisfying the equations $\left|z\right|-4=|z-i|-|z+5i|=0$, is

• A. $\sqrt{3}-i$
• B. $2\sqrt{3}-2i$
• C. $-2\sqrt{3}-2i$
• D. $0$

Answer 1

Answer: (B), (C)

The correct options are
B $-2\sqrt{3}-2i$
C $2\sqrt{3}-2i$

We have two equation $\left|z\right|-4=0$ and $|z-i|-|z+5i|=0$

Putting $z=x+iy$ i.e $x^2+y^2=16$ ....(1)

and $|x+iy-i|=|x+iy+5i|$

$\Rightarrow x^2+{(y-1)}^2=x^2+{(y+5)}^2$

$\Rightarrow y=-2$ ...(2)

Putting $y=-2$ in (1),

$x^2+4=16\Rightarrow x=\pm 2\sqrt{3}$

Hence the complex numbers $z$ satisfying the given equation are

$z_1=2\sqrt{3}-2i$ and $z_2=-2\sqrt{3}-2i$