The general 2nd degree equation

The general second degree equation in two variables is.

a*x^2+b*x*y+c*y^2+d*x+e*y+f=0

The solutions to this equation are the conic curves. In the following example by increasing the value of b, the student will be able to see that the curve passes from being a circunference to being an ellipse, a parabola (when b=0.4) and then a hyperbola (when b>0.4). 

The graphs of all the second degree equations in two variables are conic curves, although sometimes, they are degenerate as they can be a couple of straight lines, one straight line, a point, or nothing at all. The number b^2-4ac is called the discriminant of the equation and its signed determines the type of curve.

If b^2-4ac < 0 the equation is of elliptic type and its graph may be an ellipse, a circle, one point or the empty set.

If b^2-4ac = 0 the equation is of parabolic type and its graph may be a parabola, two parallel straight lines  or one straight line.

If b^2-4ac >0 the equation is of hyperbolic type and its graph may be a hyperbola or two intersecting straight lines.

The reader is welcome to verify all these statements using the above applet. For example, it can be seen that the type of curve doesn't change if only the parameters d, e and f are modified. If a and b have the same sign, then ellipses are obtained with small values of b, a single parabola with a particular value of b (b=sqrt(4ac)) and hyperbolas with large values of b. However if a and c have opposite signs the curve is always of hyperbolic type.

In order to study the degenerate cases, it is convenient to use the simplified general equation which consist in eliminating the mixed term b*x*y. The following applet shows the general equation of second degree with no  xy term. The student should do the following excercises with this applet:

Excercises.

1. Find some values of a, c, d, e and f for which the graph is a couple of intersecting straight lines.

2. Find some values of a, c, d, e and f for which the graph is a single straight line.

3. Find some values of a, c, d, e and f for which the graph is a single point.

4. Find some values of a, c, d, e and f for which the graph is the empty set.

Author: José Luis Abru León

	Ministerio de Educación, Cultura y Deporte. Año 2000