Calculus: Analytic Geometry and Calculus, with Vectors

394

Figure 6.551

Cones and conics

(6.54), we would use the formulas (2.67)
(with xo = yo = zo = 0) from Section 2.6
instead of the simpler formulas of the next
paragraph.
It is customary to say that we "rotate axes"
when, as in Figure 6.551, we introduce a sup-
plementary coordinate system that can be
obtained from the original one by rotation
about a line through the origin perpendicular to the plane of the given
coordinate system. Without going back to review formulas from Section
2.6, we shall use Figure 6.551 to derive the formulas

(6.55)

x= x' cosh -y' sin 0,

y = x' sin a +y' cos 0,

x' = x cos 6 + y sin 6

y'= -x sin0+ycos0

that relate the original and prime coordinates of a point P when the prime
coordinate system is obtained by rotating the original axes through the
angle B. The primitive formulas

x, cos 4, =sin.0, x=cos(4+6),

r r r r

sin (¢ + 0)

and the formulas for cosines and sines of sums give

x=rcos(4+6) =rcos4cos0-rsin0sin0=x'cos6-y'sin6

y = r sin (4' + B) = r cos 0 sin 0 + r sin 4, cos 0 = x' sin 0 +y' cos0.

This gives the first set of formulas (6.55). The second set can be obtained

by solving the first set for x' and y', or by making appropriate observa-

tions about the result of replacing 0 by -0.

Before attacking more ponderous expressions, we observe that the first

formulas in (6.55) show that if 0 = xy, then

(6.552) Q = (x' cos a - y' sin 0) (x' sin 6 + y' cos 0)

= (x'2 - y'2) sin 0 cos 6 + x'y'(cos2 0 - sin 2 6)

= 4(x'2 - y'2) sin 20 + x'y' cos 20.

We can eliminate the x'y' term by making cos 26 = 0 and hence by setting

20 = it/2 and 0 = ,r/4 (or 45°). This gives Q = 4(x'2 - y'2). The

graph in the xy plane of the equation xy = 1 is the same as the graph in

the prime coordinate system of the equation

(6.553)

(-X12 12

/2-)2 2)2

= 1.

Thus the graph of the equation y = 1/x is now thoroughly inducted into

the hyperbolic fraternity.

As in (6.53), let

(6.56) Q = 4x2 + 2Bxy + Cy2 + 2Dx + 2Ey + F