AMC 12 2007b Problem 24

How many pairs of positive integers $(a,b)$ are there such that $\gcd{(a,b)}=1$ and $$\frac{a}{b}+\frac{14b}{9a}$$ is an integer?

$\mathrm {(A)} 4 \mathrm {(B)} 6 \mathrm {(C)} 9 \mathrm {(D)} 12 \mathrm {(E)} \text{infinitely many}$