1975 AHSME Problem 5

The polynomial (x+y)^9 is expanded in decreasing powers of x. The second and third terms have equal values when evaluated at x=p and y=q, where p and q are positive numbers whose sum is one. What is the value of p?

\textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 1/4 \qquad \textbf{(D)}\ 3/4 \qquad \textbf{(E)}\ 8/9

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.