For z = x+iy find the relationship between x and y

[zigzag7]

Homework Statement

For z = x+iy find the relationship between x and y so that (Imz²) / z² = -i

2. The attempt at a solution
I attempted this in a few different ways (i.e. looking at the exponential and trig forms of complex numbers)... I settled on simple FOIL which gave me the following:

(x+iy)^2 = x^2 + i2xy - y^2

The imaginary part is 2xy; so:

2xy / (x^2 + i2xy - y^2) = -i <-- from original problem

from here, multiplying -i by the denominator gives:

2xy = -ix^2 + 2xy + iy^2

Cancel out 2xy to get zero on the left side, and factor out i, leaving:

x^2 = y^2

However... this does not seem to produce any solutions resulting in -i.

For example, if x=2 and y=2, z^2 = 8i ... for x=-2 and y=2, z^2 = -8i ...

The problem is, there never seems to be a sign change for Im(z^2) over z^2.

Is this problem flawed, or am I missing something obvious...?

 

[mfb]

For example, if x=2 and y=2, z^2 = 8i ... for x=-2 and y=2, z^2 = -8i ...

What is wrong with those examples?

Im(8i)=8. What is 8/(8i)?
Im(-8i)=-8. What is -8/(-8i)?

 

[zigzag7]

Ah, I didn't realize that 1/i equals -i.

Thanks for the quick response!