Question

locate the point representing the complex numbers $z$ on the Argand diagram for which
$\left|z\right|-4=|z-i|-|z+5i|=0$

Answer 1

We have two equations
$\left|z\right|-4=0$ and $\left| z-i \right| -\left| z-5i \right| =0$
Putting $z=x+iy$, these equation become
$|x+iy|=4$ i.e $x^2+y^2=16$.
And $|x+iy-i|=|x+iy+5i|$
or $x^2+(y-1{)}^2=x^2+(y+5{)}^2$
i.e,, $y=-2$
Putting $y=-2$ in $\left(1\right)$, $x^2+4=16$ or

${ x } = \pm 2\sqrt { 3 }$.
Hence the complex number $z$ satisfying the given equations are
$z_1=(2\sqrt{3},-2)$
and $z_2=(-2\sqrt{3},-2)$
that is, $z_1=-2\sqrt{3}-2i$,

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