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.
