Putnam 1994 A2

Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{1}{2} x$, the $x$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx$, the $y$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$.