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To bring the conic into standard position, the (11)
axes must be translated. Proceeding as in Example 5, we rewrite 11 as
Completing the squares yields (12)
or
If we translate the coordinate axes by means of the translation equations
then 12 becomes
which is the equation of the ellipse sketched in Figure 9.6.5. In that figure, the vectors and are the column vectors of P
—that is, the eigenvectors of A.
Figure 9.6.5
Exercise Set 9.6
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In each part, find a change of variables that reduces the quadratic form to a sum or difference of squares, and express the
1. quadratic form in terms of the new variables.
(a)
(b)
