1. ## conics

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.

2. Originally Posted by millwallcrazy
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: $x^2 + y^2 = y + x + 5$. 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... .
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