1970 AHSME Problem 35

A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served a years more, his pension would have been p dollars greater, whereas, had he served b years more b\neq a, his pension would have been q dollars greater than the original annual pension. Find his annual pension in terms of a,b,p, and q.

\textbf{(A) }\dfrac{p^2-q^2}{2(a-b)}\qquad \textbf{(B) }\dfrac{(p-q)^2}{2\sqrt{ab}}\qquad \textbf{(C) }\frac{ap^2-bq^2}{2(ap-bq)}\qquad \textbf{(D) }\frac{aq^2-bp^2}{2(bp-aq)}\qquad \textbf{(E) }\sqrt{(a-b)(p-q)}

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