Proving that an analytic function on a region is constant.

Let f=u+iv be an analytic function defined on a region D. Suppose

a*u(x,y) + b*v(x,y) = c for all x+iy in D, where a,b,c (in C) are constants, not all 0.

Prove that f is constant on D.

My guess is that I'm meant to show df/dx and df/dy = 0? Because it's constant. But I'm not sure how to go about it. I've used the Cauchy-Riemann equations to derive expressions for df/dx and df/dy with u and v, but I'm not really sure what to do with them, nor the terms of a, b and c.