University Mathematics Handbook (2015)

III. Analytic Geometry and Vectors

Chapter 3. Second-Order Curves in Plane - General Theory

(*)

is a general equation of second-order curve with two variables.

By switching the variables, that is, rotating the axis, we get

When is the solution of the equation , we can turn (*) into an equation where the coefficient of equals to zero

It is denoted the following way

After shifting the origin to point , its canonical form is obtained

Theorem: Denote

Let and be the solutions of equation

(That is, are the eigenvalues of ). Therefore:

1.  If , after rotating and shifting the axis, we obtain from equation (*) the following:

a.  If and , then the curve is an ellipse.

b.  If and , then there are no such points. The intersection is an imaginary ellipse.

c.  If and , then it is a point.

d.  If and , then the curve is a hyperbola.

e.  If and , then it is two intersecting straight lines.

2.  If and , and the equation obtained from (*) is , the curve is a parabola.

3.  If , then the equation obtained is

a.  If , then it forms two parallel straight lines.

b.  If , then it forms two coincident lines.

c.  If , then the equation holds for no point.