3.53: Distance between axis intercepts; curve intersections

Find the exact distance between the two points where the curve $9(x - 1)^{2} + 4(y - 3)^{2} = 36$ cuts the $y$-axis.

Find the coordinates of the points where the curve with equation $2x^{2} + 83xy = x^{3}y - 20x$ intersects the curve with equation $y = \dfrac{1}{x}$. Give each of your answers in the form $a + b\sqrt{c}$, where $a$ and $b$ are rational and $c$ is the smallest integer possible.