(:keywords,equation of second degree,coordinate system,change of variables:)

A general equation of second degree is of the form

Examples include, say, x^2 + (y - 2)^2 = 4 for a circle of radius 2 and center at (2, 0), parabola y = x^2, and hyperbola xy = 1. These are not just particular cases of the general equation. The general equation can be reduced to a few cases via a change of variables. To see how it is done, it is best to rewrite the equation in matrix form.

Let M = \pmatrix{A&\frac{B}{2}\\\frac{B}{2}&C} , \xi = {x \choose y} , d = {D \choose E}. Then

where T denotes *transpose* that, in this case converts 2\times 1 column vectors into 1\times 2 row vectors.

If \xi = P\xi' for some 2\times 2 matrix P, with \xi = {x' \choose y'}, then

so that the same function f(x, y) in the coordinate system x', y' becomes

where M' = P^T M P and d' = d^T P.

The purpose of this change of variables is to get rid of the Bxy term. This requires to diagonalize matrix M. To this end, we should be looking for eignevalues and eigenvectors of matrix M.

Let's consider an example taken from Brannan *et al*, p. 31.

The problem is to find a more revealing Cartesian system of coordinates for the equation

For this equation, M = \pmatrix{3&-5\\-5&3} , d = {14 \choose -2} and transform matrix

The modified equation is that of a hyperbola

**References**

- D. A. Brannan
*et al*,*Geometry*Cambridge University Press, 2002