Calculus: Analytic Geometry and Calculus, with Vectors

396 Cones and conies

(6.572) to get the information. It is of interest to observe that wecan

get information about Q without making these calculations. Even when

we do not know the numerical values of 14' and C', we know that

(6.573) B2 - f4C = -14'C'

when B' = 0 because B2 - AC is invariant under rotation of axes.In

case B2 - AC < 0, we must have A'C' > 0, and (6.572) shows that the

equation Q = 0 has elliptic type. In case B2 - AC = 0, we must have

A4'C' = 0 and the equation Q = 0 has parabolic type. In case B2 -

AC > 0, we must have A'C < 0 and the equation Q = 0 has hyperbolic

type. Sometimes it is necessary to know the fact, which was given in

Problem 32 of Section 6.4, that graphs of equations of elliptic type are

not always ellipses; they may be circles or points or empty sets. The

number (2B)2 - 4AC or 4(B2 - A4C) is called the discriminant of Q, and

its sign is the same as the sign of B2 - SIC. Hence we can put the above

results in a form that is easier to remember and use.

Theorem 6.58 The equation Q = 0 (and, in fact, Q itself) is elliptic

or parabolic or hyperbolic according as its discriminant is negative or zero

or positive.

Finally, we should know something about ways of using (6.563) to

calculate the coefficients A', B', C, D', E'. It is possible to write formulas

giving sin B and cos 0 in terms of A, B, C, but these formulas are so unat-

Figure 6.581

tractive that we shun them. Applied mathe-

maticians do not need lessons from this book to

obtain approximations to answers by finding 20

from a table or slide rule and then finding B by

gaily dividing by 2. When exact results must

be obtained, we draw a figure more or less

similar to Figure 6.581 and use elementary ideas

to obtain cos 20. When 0 _<- 9 < 7r/2, cos 0

and sin 0 must be nonnegative and their values can be calculated from

the formulas

(6.582) cos 0=

1 +

2

-cos 26,

sin B=

41 -

2

s 20

This gives the information needed for locating the prime axes and cal-

culating the prime coefficients.

Problems 6.59

1 Sketch x, y axes having origin 0 and, in the same figure, sketch primed

(or new) parallel axes with origin 0' having unprimed coordinates (2,3). Using

the primed axes, sketch the parabola having the equation y' = x'2. Find the

primed coordinates of the points 11 and B where the parabola intersects the lines

having the primed equations x' = -2 and x' = 2. Find the primed coordinates