# Conic

A conic, or conic section, is one of the two-dimensional shapes you can get by cutting a slice through a cone: an ellipse (including a circle), a parabola or a hyperbola. These shapes are also the graphs of quadratic equations in two variables.

The general Cartesian equation for these curves is $ax^2+2hxy+by^2+2fx+2gy+c=0.$ The type of curve can be determined by computing $h^2-ab$:

 $h^{2}-ab < 0$ ellipse (or circle) $h^{2}-ab = 0$ parabola $h^{2}-ab > 0$ hyperbola

There are also degenerate possibilities if the slice passes through the vertex of the cone; in terms of the equation, some choices of the parameters $a$, $b$, … will give these or equations with no real solutions.