Undergrad Need Help with this Step in a DE solution

[bigmike94]

TL;DR

There's a step in this book which i am not sure i follow

at the bottom where it says equation 6.35, the magnitude of C, how do we get from equation 6.34 to 6.35? I separated the fraction into 2, one with a real part and the other imaginary and tried taking the magnitude then and didn't come up with the same as in the book. Can someone help show me what I am missing, please?

 

[fresh_42]

This is simply the complex norm: $C=\dfrac{1}{x+iy}=\dfrac{x-iy}{(x+iy)(x-iy)}\Rightarrow |C|=\dfrac{1}{\sqrt{x^2+y^2}}=r.$