The complex number z satisfies the equation $\vert z \vert=\vert z+2\vert$.
Show that the real part is $-1$.
I know that $\vert z \vert = \sqrt{x^2+y^2}$ so I took
$\begin {align}\vert z \vert&=\vert z+2\vert \\\sqrt{x^2+y^2} &=\sqrt {x^2+y^2+2^2} \\ x^2+y^2&=x^2+y^2+2^2 \end {align}$
So after cancelling $x^2$ and $y^2$ from both sides of the equation, I am left wih $0=2^2$, which makes no sense.
How should I solve this question?
The second part of the question is as follows (which I also need help solving):
The complex number $z$ also satisfies $\vert z \vert -3=0$. Represent the two possible values of $z$ in an Argand diagram. Calculate also the two possible values of arg $z$.
Best Answer
It is not true that $|z+2|^{2}=x^{2}+y^{2}+2$. What is true is $|z+2|^{2}=|(x+2)+iy)|^{2}=(x+2)^{2}+y^{2}=x^{2}+y^{2}+2^{2}+4x$. So we get $4x+4=0$ or $x=-1$.