# Complex Number Problem #2

• Sep 29th 2013, 03:31 AM
CrispyPlanet
Complex Number Problem #2
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)?

CP
• Sep 29th 2013, 03:40 AM
Shakarri
Re: Complex Number Problem #2
No they aren't necessarily purely real or purely imaginary. z1=cai-cb, z2=a+bi. That gives z1/z2=ci
• Sep 29th 2013, 04:53 AM
CrispyPlanet
Re: Complex Number Problem #2
EDIT: I solved it for the general solution. Many thanks.
CP
• Sep 29th 2013, 04:56 PM
topsquark
Re: Complex Number Problem #2
Quote:

Originally Posted by CrispyPlanet
EDIT: I solved it for the general solution. Many thanks.
CP

Would you mind posting your solution? Someone else might benefit from seeing it. :)

-Dan
• Sep 29th 2013, 10:43 PM
CrispyPlanet
Re: Complex Number Problem #2
I let z1=a+bi, and z2=c+di. Then the modulus equality becomes: (a+c)^2 + (b+d)^2 = (a-c)^2 + (b-d)^2. I worked this out as a first step. Then take z1/z2, and work this out. You should find that it is possible to substitute what you got in step one into this division, and the outcome should be purely imaginary.

Best
CP