AHSME 1980 Problem 8

How many pairs $(a,b)$ of non-zero real numbers satisfy the equation $$\frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}$$ $\text{(A)} \ \text{none} \text{(B)} \ 1 \text{(C)} \ 2 \text{(D)} \ \text{one pair for each} ~b \neq 0$ $\text{(E)} \ \text{two pairs for each} ~b \neq 0$