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Two complex numbers z1 and z2 are taken such that |z1+z2|=|z1-z2|, and z2 not equal to zero.

Prove that z1/z2 is purely imaginary (has no real parts).

I started by taking z1=a+bi, and z2=c+di, then z1+z2=a+c+i(b+d) and z1-z2=a-c+i(b-d)

|z1+z2|=√(a+c)^2 + (b+d)^2

|z1-z2|=√(a-c)^2 + (b-d)^2

As these don't equal each other, I need to choose another two complex numbers. But I'm not sure which ones to choose. I'm assuming this is the hardest part of the problem, and having found z1 and z2 I can divide them to show that no real parts remain. Any help is much appreciated, and thank you in advance.

Prove that z1/z2 is purely imaginary (has no real parts).

I started by taking z1=a+bi, and z2=c+di, then z1+z2=a+c+i(b+d) and z1-z2=a-c+i(b-d)

|z1+z2|=√(a+c)^2 + (b+d)^2

|z1-z2|=√(a-c)^2 + (b-d)^2

As these don't equal each other, I need to choose another two complex numbers. But I'm not sure which ones to choose. I'm assuming this is the hardest part of the problem, and having found z1 and z2 I can divide them to show that no real parts remain. Any help is much appreciated, and thank you in advance.

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