Consider az + bz\' + c = 0 where z\' denotes the complex conjugate ofz and a,b,c

Question

Consider az + bz' + c = 0 where z' denotes the complex conjugate ofz and a,b,c, are complex constants. Find the conditions wherethere is a unique solution. Find the solution.

This is in the section on modulus before inequalities. Itseems like it should be simple but I am having trouble figuring outwhat to do.

I said that if z is not equal to w and az + bz' + c = 0 = aw + bw'+ c then
a(z-w) + b(z' - w') = 0 so a(z-w) = -b(z - w)' and taking the normof both sides gives. |a| = |b|.
So is it correct to say that if |a| = |b| then there is 0 or 1solution? Then what?

Explanation / Answer

Anyone?

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