The complex numbers $z$ satisfying $|z|-4=0$ and $ |z-i|-|z+5i|=0$ are ?

Question

(A) $\pm2\sqrt 3-2i$

(B) $2\sqrt 3\pm 2i$

(C) $2\pm2\sqrt 3$

(D) $\pm2-2\sqrt 3i$

Answer 1

Given: $|z|-4=0$

$\Rightarrow\:x^2+y^2=16$ where $z=x+iy$

$|z-i|-|z+5i|=0$

$\Rightarrow\:|x+i(y-1)|-|x+i(y+5)|=0$

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

$\Rightarrow\:12y+24=0$

$\Rightarrow\:y=-2$ and

$x^2=16-4=12$

$\Rightarrow\:x=\pm2\sqrt 3$

$\Rightarrow\:z=\pm2\sqrt 3-2i$