Question

Find complex numbers satisfying $\left|z\right|-4=0$ and $|z-i|-|z+5i|=0$

• A. $z_1=2\sqrt{3}-2i,z_2=-2\sqrt{3}-2i.$
• B. $z_1=2\sqrt{3}-2i,z_2=2\sqrt{3}+2i.$
• C. $z_1=2\sqrt{3}-i,z_2=-2\sqrt{3}-i.$
• D. $z_1=2\sqrt{3}-i,z_2=2\sqrt{3}+i.$

Answer 1

The correct option is A $z_1=2\sqrt{3}-2i,z_2=-2\sqrt{3}-2i.$

Let $z=x+iy$

Then, $|z|=4$

And
$\Rightarrow |z-i|=|z+5i|$

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

$\Rightarrow (2y+4)\left(6\right)=0$

$\Rightarrow 2y+4=0$

$\Rightarrow y=-2$

And
$\Rightarrow |z|=4$

$\Rightarrow x^2+y^2=16$

Now $y=-2$

Then, $x^2=16-4=12$

$\Rightarrow x=\pm 2\sqrt{3}$

Hence , $z=2\sqrt{3}-2i$ and $z=-2\sqrt{3}-2i$.