Question

Determine the complex number Z which satisfies the equations |z+3i|=|z+5-2i| and |z-4i|=|z+2i| simultaneously

Answer 1

Let $z=a+bi.$ Then

$$|z+3i|=|z+5-2i|$$ $$|a+(b+3)i|=|(a+5)+(b-2)i|$$

$$a^2+(b+3)^2=(a+5)^2+(b-2)^2$$ $$a^2+b^2+6b+9=a^2+10a+25+b^2-4b+4$$ $$10a=10b-20$$ $$a=b-2$$

$$|z-4i|=|z+2i|$$ $$|a+(b-4)i|=|a+(b+2)i|$$ $$a^2+(b-4)^2=a^2+(b+2)^2$$ $$a^2+b^2-8b+16=a^2+b^2+4b+4$$ $$12b=12$$ $$b=1$$

$$a=1-2=-1$$

$z=-1+i$