Calculus: Analytic Geometry and Calculus, with Vectors

6.5 Translation and rotation of axes 395

and suppose that .4, B, C are not all zero. Rotating the x, y axes through
the angle 0 gives new coordinates x', y', and using (6.55) shows that, in
the new coordinates,

(6.561) Q = 4'x'2 + 2B'x'y' + C'y'2 + 2D'x' + 2E'y' + F

where A' = .4 cost 0 + 2B sin 0 cos 0 + C sine 6

B' = (C - 4) sin 0 cos 0 + B(cos2 0 - sine 0)

C' = -4 sin2 0 - 2B sin 0 cos B + C cos2 0

D' =Dcos0+Bsin0

E' = -Dsin0+Ecos0.

Considerable information can be extracted from the above formulas

without the aid of noncranial electronics. For each B, we obtain the first

of the formulas

(6.562) A'+C' =-4 + C, =B2-SIC

by adding the expressions for A' and C'. To prove the second formula,

we can write an expression for the left side, cancel the terms that cancel,

and obtain the result with the aid of the identity

cos4 0 + 2 cost 0 sin2 0 + sin4 0 = (cos' 0 + sin2 0)2 = 1.

Because of (6.562), the quantities fl + C and B2 - 4C are said to be

invariant under rotation of axes.

The basic importance of the formulas for A', B', C', D', E' lies in the

fact that they show us how to determine 0 and the coefficients in (6.561)

so that B' is zero and the objectionable term is missing. When we write

the formula for B' in the form

B' =-(C-A) sin 20 + B cos 20,

we can see that it is easy to find a unique angle 0 for which 0 <_ 0 <x/2

and B' = 0. In case C = A, we set 0 = 1r/2. In case C .4, we choose

20 such that 0 < 26 < 7r and

(6.563) tan 20 -

A2B

C

Without repeating the formulas for the prime coefficients and the method

by which 0 is found, we put our main result in the following theorem.

Theorem 6.57 Each 0 of the form

(6.571) Q = Axe + 2Bxy + Cy2 + 2Dx + 2Ey + F

can be put in the form

(6.572) Q = A'x'2 + C'y'2 + 2D'x + 2E'y + F

by suitable choice of an angle 8 for which 0 < 0 < a/2.

When we are asked about the graph of Q = 0, we can calculate the

coefficients in (6.572) and then (completing squares when necessary) use