ABSTRACT
We now classify the curves given by a general polynomial equation of the second degree. We prove that such a general quadratic equation can be re duced to a canonical form of equation. At the same time we show how this canonical form can be obtained. The method of obtaining the canonical form involves rotations and translations of the coordinate axes, or equiva lent^ of the plane, and enables us to determine whether the given quadratic equation is the equation of an ellipse, a hyperbola, or a parabola, and to find information about its axes, semi-axes, foci, asymptotes, etc. We give two methods for diagonalising the quadratic terms, namely the geometric method and the algebraic method. For those familiar with the diagonalisation of simple quadratic forms the algebraic method can give the results more rapidly and is recommended. However the examples and exercises considered in this chapter can also be solved using the geometric method. A resumé of the theory of matrices and determinants used in the algebraic method of diagonalising quadratic forms is included in §2.2 for reference.
