Calculus: Analytic Geometry and Calculus, with Vectors

382 Cones and conics

and for other quadrants we use symmetry. We can use the Pythagoras

theorem to calculate the distance from the center to a focus if we remem-

ber Figure 6.48, a simplified version of Figure 6.44, which shows that if we

fix one point of a compass at the origin, then the circle through the corners

of the box meets the principal axis at the foci. If we remember that this

distance is ae (as it also is for ellipses), then we can calculate the eccen-

tricity e. Finally, if we remember that the distance from the center to a

directrix is ale (as it also is for ellipses), we can calculate this distance.

The numbers ae and ale are still the key numbers.

Problems 6.49

1 For each of the following pairs of values of a and b, sketch the hyperbola

having the equation

xz ys

a2 b2-1,

find the eccentricity, find the foci (give coordinates), find the directrices (give

equations), and find the asymptotes (give equations). Try to cultivate the

ability to use the Pythagoras theorem and key numbers without use of books or

notes. Check numerical results by use of the fact that the distance p from a

focus to its directrix must satisfy the equation e2p2 = b2(e2 - 1).

(a) a=5,b=2 (b) a=2,b=5 (c) a=b=1

2 The equation

y2 x2

22-32=1

differs from equations of hyperbolas having their principal axes and foci on the

x axis because the roles of x and y are interchanged. Nevertheless, plot the

points on the graph obtained by setting x = 0 and then y = 0 and then draw the

helpful box and sketch the hyperbola. Then proceed to find the eccentricity,

foci, directrices, and asymptotes. Repeat the process when 2 and 3 are respec-

tively replaced by

(a) 2 and 5 (b) 5 and 2 (c) 1 and 1

Remark: The graphs of the equations

X2s s 2 2

b2a21

are hyperbolas having the same helpful box and the same asymptotes. Each

hyperbola is said to be the conjugate of the other.

3 Sketch graphs of the equations

(a) (x

521) -

(Y 222)

= 1 (b) 222)2-(x

521)2

= 1

Remark: Good clean starts are made by setting y = 2 orx = 1 and remembering

that squares of our numbers (which are always real numbers)are never negative.