Complex Number Proof

[Makatak]

I've got this question:

Let z=x+iy and z[sub:3p0lrmpq]3[/sub:3p0lrmpq]=the conjugate of z[sub:3p0lrmpq]1[/sub:3p0lrmpq] . Show that if |z[sub:3p0lrmpq]1[/sub:3p0lrmpq]-z[sub:3p0lrmpq]2[/sub:3p0lrmpq]|/|1-z[sub:3p0lrmpq]3[/sub:3p0lrmpq]z[sub:3p0lrmpq]2[/sub:3p0lrmpq]|=1 , |z[sub:3p0lrmpq]1[/sub:3p0lrmpq]|=1 or |z[sub:3p0lrmpq]2[/sub:3p0lrmpq]|=1

I start by making |z[sub:3p0lrmpq]1[/sub:3p0lrmpq]-z[sub:3p0lrmpq]2[/sub:3p0lrmpq]|=|1-z[sub:3p0lrmpq]3[/sub:3p0lrmpq]z[sub:3p0lrmpq]2[/sub:3p0lrmpq]|, but i'm not sure if i'm on the right track.
I tried converting all the z's to x's and y's but that got messy. I'm really confused as to how I am meant to do this question, especially the modulus parts.

 

[DrMike]

I've got this question:

Let z=x+iy and z[sub:k31i9xot]3[/sub:k31i9xot]=the conjugate of z[sub:k31i9xot]1[/sub:k31i9xot] . Show that if |z[sub:k31i9xot]1[/sub:k31i9xot]-z[sub:k31i9xot]2[/sub:k31i9xot]|/|1-z[sub:k31i9xot]3[/sub:k31i9xot]z[sub:k31i9xot]2[/sub:k31i9xot]|=1 , |z[sub:k31i9xot]1[/sub:k31i9xot]|=1 or |z[sub:k31i9xot]2[/sub:k31i9xot]|=1

I start by making |z[sub:k31i9xot]1[/sub:k31i9xot]-z[sub:k31i9xot]2[/sub:k31i9xot]|=|1-z[sub:k31i9xot]3[/sub:k31i9xot]z[sub:k31i9xot]2[/sub:k31i9xot]|, but i'm not sure if i'm on the right track.
I tried converting all the z's to x's and y's but that got messy. I'm really confused as to how I am meant to do this question, especially the modulus parts.

It's probably a good idea to do as much simplification in complex numbers as you can, before putting in the x's and y's. Also, remember that \(\displaystyle |z|^2=z{\bar z}\), so you can get rid of the |.| signs. Start by saying |z[sub:k31i9xot]1[/sub:k31i9xot]-z[sub:k31i9xot]2[/sub:k31i9xot]|=|1-z[sub:k31i9xot]3[/sub:k31i9xot]z[sub:k31i9xot]2[/sub:k31i9xot]| => |z[sub:k31i9xot]1[/sub:k31i9xot]-z[sub:k31i9xot]2[/sub:k31i9xot]|[sup:k31i9xot]2[/sup:k31i9xot]=|1-z[sub:k31i9xot]3[/sub:k31i9xot]z[sub:k31i9xot]2[/sub:k31i9xot]|[sup:k31i9xot]2[/sup:k31i9xot], plug in your z[sub:k31i9xot]3[/sub:k31i9xot], get rid of the |.| signs, and see what cancels/factorizes.