# Find all complex numbers satisfying |z+3+4i|=|z-5|

Find all complex numbers satisfying
$|z+3+4i|=|z-5|$
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Step 1
$|z+3+4i|=|z-5|$
Let $z=z+iy$
So $|\left(x+3\right)+i\left(y+4\right)|=|\left(x-5\right)+iy|$
$\sqrt{{\left(x+3\right)}^{2}+{\left(y+4\right)}^{2}}=\sqrt{{\left(x-5\right)}^{2}+{y}^{2}}$
${x}^{2}+9+6x+{y}^{2}+16+8y={x}^{2}+25-10x+{y}^{2}$
$16x+8y=0$
$wx+y=0$
$y=-2x$
So $z=x-2i$
$z=x\left(1-2i\right)$
Ex $x=1⇒z=1-2i$
$x=2⇒z=2-4i$
$z=3⇒z=3-6i$