James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Chapter 10
Concept Check Answers (continued)
8. (a) Give a definition of a hyperbola in terms of foci.
A hyperbola is a set of points in a plane the difference
of whose distances from two fixed points (the foci) is
a constant. This difference should be interpreted as the
larger distance minus the smaller distance.
(b) Write an equation for the hyperbola with foci s6c, 0d and
vertices s6a, 0d.
x 2
a 2 y 2
2 b − 1 2
where c 2 − a 2 1 b 2 .
(c) Write equations for the asymptotes of the hyperbola in
part (b).
y − 6 b a x
9. (a) What is the eccentricity of a conic section?
If a conic section has focus F and corresponding directrix
l, then the eccentricity e is the fixed ratio | PF |y| Pl | for
points P of the conic section.
(b) What can you say about the eccentricity if the conic
section is an ellipse? A hyperbola? A parabola?
e , 1 for an ellipse; e . 1 for a hyperbola; e − 1 for a
parabola
(c) Write a polar equation for a conic section with eccentricity
e and directrix x − d. What if the directrix is
x − 2d? y − d? y − 2d?
directrix x − d: r −
x − 2d: r −
y − d: r −
y − 2d: r −
ed
1 1 e cos
ed
1 2 e cos
ed
1 1 e sin
ed
1 2 e sin
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