1985 AHSME Problem 19

Consider the graphs y=Ax^2 and y^2+3=x^2+4y, where A is a positive constant and x and y are real variables. In how many points do the two graphs intersect?

\mathrm{(A) \ }\text{exactly }4 \qquad \mathrm{(B) \ }\text{exactly }2 \qquad

\mathrm{(C) \ }\text{at least }1,\text{ but the number varies for different positive values of }A \qquad

\mathrm{(D) \ }0\text{ for at least one positive value of }A \qquad \mathrm{(E) \ }\text{none of these}

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