Calculus: Analytic Geometry and Calculus, with Vectors

6.5 Translation and rotation of axes 393

or, equivalently,

(6.512) x=x'+5,

the equation takes the simpler form

y = y' + 4,

(6.513) x'2 + y'2 = 9.

If we think of x' and y' as being coordinates (the prime or primed coor-

dinates), then (6.513) looks like the equation of a circle havingits center

at the originof the new prime coordinate system of Figure 6.52. It is

customary to say that we"translate axes" when, as in Figure 6.52, we

introduce a supplementary coordinate system that can be obtained from

the original one by translations, that is, by slidings free from rotations.

It sometimes happens that, in more complicated situations, introduction

of a supplementary coordinate system helps us to determine the nature

and position of the graph of a given equation.

We now begin consideration of the graph of the equation Q = 0, where

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

it being supposed that A, B, C, D, B, F are given constants for which

A, B, C are not all 0. In case B = 0, the following results can be obtained

more quickly by completing squares. In any case, we undertake to

determine constants h and k such that the substitution x = x' + h,

y = y' + k will yield a simpler expression for Q. Substitution gives

(6.531) Q = Ix" + 2Bx'y' + Cy'2 + 2D'x' + 2E'y' + F'

where D' = Ah + Bk + D

E' =Bh+Ck+E

F = Bh2 + 2Bhk + Ck2 + 2Dh + 2Ek + F.

In case B2 - AC 0 0, we can simplify the expression for Q in (6.531) by

determining h and k so that D' = E' = 0. Except in the special case

in which F = 0 when h and k are so determined that D' = E' = 0, the

equation Q = 0 resulting from making D' = E' = 0 is not easily graphed

unless A = C = 0 or B = 0. Hence, at least in the study of Q when

B s 0, the results that lie ahead are much more important than those

obtained by translation of axes. For those who may become interested

in such things, it may be remarked that the quadratic form 0 in

(6.54) Q = Axx + Bxy + Dxz

+ Byx + Cyy + Eyz

+Dzx+Ezy+Fzz

reduces to (6.53) when z = 1 and that there are places in pure and

applied mathematics where these things are important. To attack