UNsolve 2 variable quadratic equations & inequalities

The solution set of a quadratic equation in 2 variables, f(x,y) = 0, is a curve in the x,y plane. The solution set of a quadratic inequality in 2 variables f(x,y) > 0 is EITHER all the points in the region in the x,y plane enclosed by the curve, OR all the points outside the curve. To UNsolve a quadratic equation or inequality, use the a, b and z0 sliders in the upper panel to fix the solution set. The left panel will show the solution set. The right hand panel will show you a quadratic equation or inequality that has that solution set. You can use the A slider in the upper panel to see other quadratic equations or inequalities that have the same solution set. For a given solution set, how many equations or inequalities are there that have that solution set? How do you know? Can you prove it?