Given that the parabolas
x^2 = y + 2
y^2 = x + 3
meet in four points, show that these four points lie on a circle, and also on a
rectangular hyperbola.
Add the two equations together: . Any point that satisfies the original two equations will also satisfy that new equation. But that new equation is the equation of a circle. So any point that lies on both parabolas also lies on the circle.
If you subtract the equations instead of adding them, you'll get the equation of a rectangular hyperbola... .