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I have the following equation:

$$(z+\bar z)|w|^2 - (z-\bar z)|w|i - 2(z + \bar z) = 0. $$

I need to show that $z$ can't be an imaginary number.

then that the image of $z$ passes from (0,0)

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by $z^*$ do you mean the complex conjugate? then it is better to use $\bar z$ – Dennis Gulko Oct 28 '12 at 14:52
Yes, that's what I mean. – Chris Oct 28 '12 at 14:53
up vote 1 down vote accepted

By rearranging you equation, we get $$(z+\bar z)|w|^2 - 2(z + \bar z) = (z-\bar z)|w|i$$ Do you know that $z+\bar z=2\Re(z)$ and $z-\bar z=2i\Im(z)$ - if you never seen this before, prove it by setting $z=x+iy$ and computing directly. So you have: $$2(|w|^2-2)\Re(z)=-2|w|\Im(z)$$ What happens if $z$ is imaginary?

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