Page 735 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution

Rearranging terms gives
Completing the squares yields
or
or

Translating the axes by means of the translation equations
yields

which is the equation of a hyperboloid of one sheet.

Eliminating Cross-Product Terms

The procedure for identifying quadrics that are rotated out of standard position is similar to the procedure for conics. Let Q
be a quadric surface whose equation in -coordinates is

We want to rotate the -coordinate axes so that the equation of the quadric in the new (2)
cross-product terms. This can be done as follows: -coordinate system has no

Step 1. Find a matrix P that orthogonally diagonalizes .

Step 2. Interchange two columns of P, if necessary, to make det . This ensures that the orthogonal coordinate
transformation

is a rotation. (3)
-coordinates with no cross-product
Step 3. Substitute 3 into 2. This will produce an equation for the quadric in
terms. (The proof is similar to that for conics and is left as an exercise.)

The following theorem summarizes this discussion.