Page 724 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

By completing the squares* on the two expressions in parentheses, we obtain (3)
or

If we translate the coordinate axes by means of the translation equations
then 3 becomes
which is the equation of an ellipse in standard position in the -system. This ellipse is sketched in Figure 9.6.3.

Figure 9.6.3

Eliminating the Cross-Product Term matrices,

We shall now show how to identify conics that are rotated out of standard position. If we omit the brackets on
then 2 can be written in the matrix form

or
where

Now consider a conic C whose equation in -coordinates is

We would like to rotate the -coordinate axes so that the equation of the conic in the new (4)
cross-product term. This can be done as follows. -coordinate system has no

Step 1. Find a matrix