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).

So I said there are 4 possibilities (ie z1=a or bi, and z2=bi or a), and worked out whether the modulus equality holds for each (it does). The corresponding theta for z1 and z2 is then either 0 or pi/2. Then:

z1/z2 = a/bi=-(a/b)*i

or

z1/z2=(b/a)*i

Is this about right? For instance I'm really not certain about the angles. If z1=a (ie real only), then its theta is 0 right? And if z2=bi (imaginary only), it's theta is (pi/2)?

Many thanks for your help in advance.

CP