Section 22–5
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The Hyperbola
y
Light ray
240 km
F
F′
O
P
Shore
x
120 km
x
18.0 mm
26.0 mm
y
FIGURE 22–85
FIGURE 22–84 Hyperbolic mirror. This type of mirror
is used in the Cassegrain form of reflecting telescope.
Case Study Discussion—Parabolic Microphone
The parabolic microphone can be made from inexpensive materials. A few sites on the
Internet describe “dollar store” projects that use a small umbrella hat to form the reflector
dish. We don’t have room for a full set of instructions here, but you will be able to vary
the position of the microphone to see just how well the reflector works.
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CHAPTER 22 REVIEW PROBLEMS
x
Q
0
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1. Find the distance between the points (3,0) and (7,0).
2. Find the distance between the points (4, 4) and (1, 7).
3. Find the slope of the line perpendicular to a line that has an angle of inclination of
34.8.
4. Find the angle of inclination in degrees of a line with a slope of 3.
5. Find the angle of inclination of a line perpendicular to a line having a slope of 1.55.
6. Find the angle of inclination of a line passing through (3, 5) and (5, 6).
7. Find the slope of a line perpendicular to a line having a slope of a/2b.
8. Write the equation of a line having a slope of 2 and a y intercept of 5.
9. Find the slope and y intercept of the line 2y 5 3(x 4).
10. Write the equation of a line having a slope of 2p and a y intercept of p 3q.
11. Write the equation of the line passing through (5, 1) and (2, 6).
12. Write the equation of the line passing through (r, s) and (2r, s).
13. Write the equation of the line having a slope of 5 and passing through the point (4, 7).
14. Write the equation of the line having a slope of 3c and passing through the point (2c, c 1).
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Chapter 22
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Analytic Geometry
15. Write the equation of the line having an x intercept of 3 and a y intercept of 7.
16. Find the acute angle between two lines if one line has a slope of 1.50 and the other has a
slope of 3.40.
17. Write the equation of the line that passes through (2, 5) and is parallel to the x axis.
18. Find the angle of intersection between line L1 having a slope of 2 and line L2 having a slope
of 7.
19. Find the directed distance AB between the points A(2, 0) and B(5, 0).
20. Find the angle of intersection between line L1 having an angle of inclination of 18 and line
L2 having an angle of inclination of 75.
21. Write the equation of the line that passes through (3, 6) and is parallel to the y axis.
22. Find the increments in the coordinates of a particle that moves along a curve from (3, 4) to
(5, 5).
23. Find the area of a triangle with vertices at (6, 4), (5, 2), and (3, 4).
In problems 24 through 33, a tangent T, of slope m, and a normal N are drawn to a curve at the
point P(x1, y1), as shown in Fig. 22–86. Show the following:
y
D
T
N
m
P(x1, y1)
A
O
C
x
G
B
FIGURE 22–86
Tangent and normal to a curve.
24. The equation of the tangent is
y y1 m(x x1)
25. The equation of the normal is
x x1 m(y y1) 0
26. The x intercept A of the tangent is
x1 y1/m
27. The y intercept B of the tangent is
y1 mx1
28. The length of the tangent from P to the x axis is
y1 PA 1 m2
m
29. The length of the tangent from P to the y axis is
PB x1 1 m2
Section 22–5
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The Hyperbola
30. The x intercept C of the normal is
31. The y intercept D of the normal is
x1 my1
y1 x1m
31. The length of the normal from P to the x axis is
PC = y1 1 m2
33. The length of the normal from P to the y axis is
x1 PD = 1 m2
m
Identify the curve represented by each equation. Find, where applicable, the centre, vertices,
foci, radius, semiaxes, and so on.
34. x2 2x 4y2 16y 19
35. x2 6x 4y 3
36. x2 y2 8y
37. 25x2 200x 9y2 90y 275
38. 16x2 9y2 144
39. x2 y2 9
40. Write the equation for an ellipse whose centre is at the origin, whose major axis (2a) is 20
and is horizontal, and whose minor axis (2b) equals the distance (2c) between the foci.
41. Write an equation for the circle passing through (0, 0), (8, 0), and (0, 6).
42. Write the equation for a parabola whose vertex is at the origin and whose focus is (4.25, 0).
43. Write an equation for a hyperbola whose transverse axis is horizontal, with centre at
(1, 1) passing through (6, 2) and (3, 1).
44. Write the equation of a circle whose centre is (5, 0) and whose radius is 5.
45. Write the equation of a hyperbola whose centre is at the origin, whose transverse axis 8 and is horizontal, and passing through (10, 25).
46. Write the equation of an ellipse whose foci are (2, 1) and (6, 1), and the sum of the focal
radii is 10.
47. Write the equation of a parabola whose axis is the line y 7, whose vertex is 3 units to
the right of the y axis, and passing through (4, 5).
48. Find the intercepts of the curve y2 4x 6y 16.
49. Find the points of intersection of x2 y2 2x 2y 2 and
3x2 3y2 5x 5y 10.
50. A stone bridge arch is in the shape of half an ellipse and is 15.0 m wide and 5.00 m high.
Find the height of the arch at a distance of 6.00 m from its centre.
51. Write the equation of a hyperbola centred at the origin, where the conjugate axis 12 and
is vertical, and the distance between foci is 13.
52. A parabolic arch is 5.00 m high and 6.00 m wide at the base. Find the width of the arch at
a height of 2.00 m above the base.
53. A stone thrown in the air follows a parabolic path and reaches a maximum height of 56.0 ft.
in a horizontal distance of 48.0 ft. At what horizontal distances from the launch point will
the height be 25.0 ft.?
Writing
54. Write a short paragraph explaining, in your own words, what is meant by the slope of a
curve.
55. Suppose that the day before your visit to your former high school math class, the teacher
unexpectedly asks you to explain how that day’s topic, the conic sections, got that name.
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Chapter 22
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Analytic Geometry
Write a paragraph on what you will tell the class about how the circle, ellipse, parabola,
and hyperbola (and the point and straight line, too) can be formed by intersecting a plane
and a cone. You may plan to illustrate your talk with a clay model or a piece of cardboard
rolled into a cone. You will probably be asked what the conic sections are good for, so
write out at least one use for each curve.
Team Projects
56. There are an infinite number of number pairs p and q (such as 3 and 1.5) whose product
equals their sum. Write an expression for q as a function of p, and make a graph of p versus
q for all of the real numbers, not just integer values, from 5 to 5. With a suitable shift of
axes, show that the graph is a hyperbola of the form y 1x.
57. We have already shown how to graph each of the conic sections. Now graph, in the same
viewing window, the ellipse
2
x2 y
=1
a2 b2
the hyperbola
2
x2 y
=1
a2 b2
and the hyperbola
y2 x2
=1
a2 b2
using the same values of a and b for each. Also graph the asymptotes of the hyperbolas.
58. If the area of an ellipse is twice the area of the inscribed circle, find the length of the semimajor axis of the ellipse.
59. If a projectile is launched with a horizontal velocity vx and vertical velocity vy, its position
after t seconds is given by
x vxt
y vyt (g/2)t2
where g is the acceleration due to gravity. Eliminate t from this pair of equations to get
y f(x), and show that this equation represents a parabola.