Correlation of combined variables

Dear all

I have got the following problem:

Given are two correlated, normally distributed variables A and B and corresponding variances and correlation. Both means are zero.

Now each variable is combined (with different weights a and b) with an additional normally distributed variable C, which has zero mean as well, known variance and is not correlated with A or B. That means:

A_combined = a*A + (1-a)*C

B_combined = b*B + (1-b)*C

Is there any way to analytically derive the correlation between A_combined and B_combined? The solution for equal weights a=b is quite straight forward, but I really got my wires crossed for a<>b...

Thank you very much for your help!

Marc