616
9 Additional Topics in Analytic Geometry
53. Space Science. A designer of a 200-foot-diameter parabolic electromagnetic antenna for tracking space probes
wants to place the focus 100 feet above the vertex (see the
figure).
(A) Find the equation of the parabola using the axis of the
parabola as the y axis (up positive) and vertex at the
origin.
(B) Determine the depth of the parabolic reflector.
54. Signal Light. A signal light on a ship is a spotlight with parallel reflected light rays (see the figure). Suppose the parabolic reflector is 12 inches in diameter and the light source
is located at the focus, which is 1.5 inches from the vertex.
Signal light
200 ft
Focus
Focus
100 ft
Radiotelescope
(A) Find the equation of the parabola using the axis of the
parabola as the x axis (right positive) and vertex at the
origin.
(B) Determine the depth of the parabolic reflector.
SECTION
9-2
Ellipse
•
•
•
•
Definition of an Ellipse
Drawing an Ellipse
Standard Equations and Their Graphs
Applications
We start our discussion of the ellipse with a coordinate-free definition. Using this definition, we show how an ellipse can be drawn and we derive standard equations for
ellipses specially located in a rectangular coordinate system.
• Definition of
The following is a coordinate-free definition of an ellipse:
an Ellipse
DEFINITION 1
Ellipse
An ellipse is the set of all points P in a plane such that the sum of the distances
of P from two fixed points in the plane is constant. Each of the fixed points, F
and F, is called a focus, and together they are called foci. Referring to the figure, the line segment VV through the foci is the major axis. The perpendicular bisector BB of the major axis is the minor axis. Each end of the major axis,
9-2
Ellipse
617
V and V, is called a vertex. The midpoint of the line segment FF is called the
center of the ellipse.
d1 d2 Constant
B
V
d1
P
F
d2
F
V
B
• Drawing an Ellipse
An ellipse is easy to draw. All you need is a piece of string, two thumbtacks, and a
pencil or pen (see Fig. 1). Place the two thumbtacks in a piece of cardboard. These
form the foci of the ellipse. Take a piece of string longer than the distance between
the two thumbtacks—this represents the constant in the definition—and tie each end
to a thumbtack. Finally, catch the tip of a pencil under the string and move it while
keeping the string taut. The resulting figure is by definition an ellipse. Ellipses of different shapes result, depending on the placement of thumbtacks and the length of the
string joining them.
FIGURE 1 Drawing an ellipse.
Note that d 1 d 2 always
adds up to the length of the
string, which does not change.
P
d2
d1
String
Focus
• Standard Equations
and Their Graphs
Focus
Using the definition of an ellipse and the distance-between-two-points formula, we
can derive standard equations for an ellipse located in a rectangular coordinate system. We start by placing an ellipse in the coordinate system with the foci on the x
axis equidistant from the origin at F(c, 0) and F(c, 0), as in Figure 2.
y
FIGURE 2 Ellipse with foci on x
axis.
P(x, y)
d1
F
(c, 0)
d2
0
x
F(c, 0)
d1 d2 Constant
c0
618
9 Additional Topics in Analytic Geometry
For reasons that will become clear soon, it is convenient to represent the constant sum d1 d2 by 2a, a 0. Also, the geometric fact that the sum of the lengths
of any two sides of a triangle must be greater than the third side can be applied to
Figure 2 to derive the following useful result:
d(F, P) d(P, F) d(F, F)
d1 d2 2c
2a 2c
ac
(1)
We will use this result in the derivation of the equation of an ellipse, which we now
begin.
Referring to Figure 2, the point P(x, y) is on the ellipse if and only if
d1 d 2 2a
d(P, F
) d(P, F) 2a
(x c)2 (y 0)2 (x c)2 (y 0)2 2a
After eliminating radicals and simplifying, a good exercise for you, we obtain
(a 2 c 2)x 2 a 2y 2 a 2(a 2 c 2)
2
(2)
2
x
y
2
1
2
a
a c2
(3)
Dividing both sides of equation (2) by a2(a2 c2) is permitted, since neither a2 nor
a2 c2 is 0. From equation (1), a c; thus a2 c2 and a2 c2 0. The constant
a was chosen positive at the beginning.
To simplify equation (3) further, we let
b2 a2 c2
b0
(4)
to obtain
x2 y2
1
a2 b2
(5)
From equation (5) we see that the x intercepts are x a and the y intercepts are
y b. The x intercepts are also the vertices. Thus,
Major axis length 2a
Minor axis length 2b
To see that the major axis is longer than the minor axis, we show that 2a 2b.
Returning to equation (4),
b2 a2 c2
b c a
2
2
2
a, b, c 0
9-2
b2 a2
Ellipse
619
Definition of b2 a2 0
(b a)(b a) 0
ba0
Since b a is positive, b a must be negative.
ba
2b 2a
2a 2b
Length of
Length of
major
axis minor axis If we start with the foci on the y axis at F(0, c) and F(0, c) as in Figure 3,
instead of on the x axis as in Figure 2, then, following arguments similar to those
used for the first derivation, we obtain
x2 y2
1
b2 a2
ab
(6)
where the relationship among a, b, and c remains the same as before:
b2 a2 c2
(7)
The center is still at the origin, but the major axis is now along the y axis and the
minor axis is along the x axis.
y
FIGURE 3 Ellipse with foci on y
axis.
F(0, c)
d1
P(x, y)
x
0
d2
F
(0, c)
d1 d2 Constant
c0
To sketch graphs of equations of the form (5) or (6) is an easy matter. We find
the x and y intercepts and sketch in an appropriate ellipse. Since replacing x with x,
or y with y produces an equivalent equation, we conclude that the graphs are symmetric with respect to the x axis, y axis, and origin. If further accuracy is required,
additional points can be found with the aid of a calculator and the use of symmetry
properties.
Given an equation of the form (5) or (6), how can we find the coordinates of the
foci without memorizing or looking up the relation b2 a2 c2? There is a simple
geometric relationship in an ellipse that enables us to get the same result using the
Pythagorean theorem. To see this relationship, refer to Figure 4(a). Then, using the
620
9 Additional Topics in Analytic Geometry
definition of an ellipse and 2a for the constant sum, as we did in deriving the standard equations, we see that
d d 2a
2d 2a
d a2
Thus:
The length of the line segment from the end of a minor axis to a focus
is the same as half the length of a major axis.
This geometric relationship is illustrated in Figure 4(b). Using the Pythagorean theorem for the triangle in Figure 4(b), we have
b2 c2 a2
or
b2 a2 c2
Equations (4) and (7)
or
c2 a2 b2
Useful for finding the foci, given a and b
Thus, we can find the foci of an ellipse given the intercepts a and b simply by using
the triangle in Figure 4(b) and the Pythagorean theorem.
y
FIGURE 4 Geometric relationships.
b
d
F
c
a
y
x2
y2
21
2
a
b
ab0
b
d
0
a2 b2 c2
a
F
c
a
x
a
c
0
c
a
b
b
(a)
(b)
We summarize all of these results for convenient reference in Theorem 1.
Theorem 1
Standard Equations of an Ellipse with Center at (0, 0)
1.
x2 y2
1
a2 b2
ab0
x
9-2
x intercepts: a (vertices)
y intercepts: b
Foci: F(c, 0), F(c, 0)
y
b
c2 a2 b2
Major axis length 2a
Minor axis length 2b
621
Ellipse
a
a
F
c
F
c
0
a
x
b
2.
x2 y2
1
b2 a2
ab0
y
a
x intercepts: b
y intercepts: a (vertices)
Foci: F(0, c), F(0, c)
c F
a
c a b
2
2
2
Major axis length 2a
Minor axis length 2b
b
EXAMPLE 1
b
x
c F
[Note: Both graphs are symmetric
with respect to the x axis, y axis,
and origin. Also, the major axis is
always longer than the minor axis.]
EXPLORE-DISCUSS 1
0
a
The line through a focus F of an ellipse that is perpendicular to the major axis
intersects the ellipse in two points G and H. For each of the two standard equations of an ellipse with center (0, 0), find an expression in terms of a and b for
the distance from G to H.
Graphing Ellipses
Sketch the graph of each equation, find the coordinates of the foci, and find the lengths
of the major and minor axes.
(A) 9x2 16y2 144
Solutions
(B) 2x2 y2 10
(A) First, write the equation in standard form by dividing both sides by 144:
9x2 16y2 144
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16y2 144
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x2
y2
1
16
9
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a 2 16 and b 2 9
622
9 Additional Topics in Analytic Geometry
Locate the intercepts:
x intercepts: 4
y intercepts: 3
and sketch in the ellipse, as shown in Figure 5.
y
Foci: c2 a2 b2
3
4
4
F
c
0
F
c
16 9
4
x
7
c 7
3
FIGURE 5 9x2 16y2 144.
c is positive
Thus, the foci are F(7, 0) and F(7, 0).
Major axis length 2(4) 8
Minor axis length 2(3) 6
(B) Write the equation in standard form by dividing both sides by 10:
2x2 y2 10
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y2
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x2
y2
1
5
10
a 2 10 and b 2 5
Locate the intercepts:
x intercepts: 5 2.24
y intercepts: 10 3.16
and sketch in the ellipse, as shown in Figure 6.
y
Foci: c2 a2 b2
10
c F
5
0
10
5
10 5
5
x
c 5
c F
10
FIGURE 6 2x2 y2 10.
Thus, the foci are F(0, 5) and F(0, 5).
Major axis length 210 6.32
Minor axis length 25 4.47
To graph the equation 9x2 16y2 144 of Example 1A on a graphing
144 9x2
utility we first solve the equation for y, obtaining y . We then graph
16
Remark.
9-2
each of the two functions. The graph of y ellipse, and the graph of y Matched Problem 1
623
144 9x2
is the upper half of the
16
144 9x
is the lower half.
16
2
Sketch the graph of each equation, find the coordinates of the foci, and find the lengths
of the major and minor axes.
(A) x2 4y2 4
EXAMPLE 2
Ellipse
(B) 3x2 y2 18
Finding the Equation of an Ellipse
Find an equation of an ellipse in the form
x2 y2
1
M N
M, N 0
if the center is at the origin, the major axis is along the y axis, and:
(A) Length of major axis 20
Length of minor axis 12
(B) Length of major axis 10
Distance of foci from center 4
y
Solutions
10
10
10
(A) Compute x and y intercepts and make a rough sketch of the ellipse, as shown in
Figure 7.
x2 y2
1
b2 a2
x
a
10
FIGURE 7
20
10
2
b
12
6
2
x2
y2
1
36 100
x2
y2
1.
36 100
(B) Make a rough sketch of the ellipse, as shown in Figure 8; locate the foci and y
intercepts, then determine the x intercepts using the special triangle relationship
discussed earlier.
y
x2 y2
1
b2 a2
5
4
5
b
0
b
x
a
FIGURE 8
y2
x2
1.
9
25
b2 52 42 25 16 9
b3
2
5
10
5
2
2
x
y
1
9
25
624
9 Additional Topics in Analytic Geometry
Matched Problem 2
Find an equation of an ellipse in the form
x2 y2
1
M N
M, N 0
if the center is at the origin, the major axis is along the x axis, and:
(A) Length of major axis 50
Length of minor axis 30
EXPLORE-DISCUSS 2
(B) Length of minor axis 16
Distance of foci from center 6
Consider the graph of an equation in the variables x and y. The equation of its
magnification by a factor k 0 is obtained by replacing x and y in the equation
by x/k and y/k, respectively.
(A) Find the equation of the magnification by a factor 3 of the ellipse with equation (x2/4) y2 1. Graph both equations.
(B) Give an example of an ellipse with center (0, 0) with a b that is not a magnification of (x2/4) y2 1.
(C) Find the equations of all ellipses that are magnifications of (x2/4) y2 1.
• Applications
You are no doubt aware of many occurrences and uses of elliptical forms: orbits of
satellites, planets, and comets; shapes of galaxies; gears and cams; some airplane
wings, boat keels, and rudders; tabletops; public fountains; and domes in buildings
are a few examples (see Fig. 9). A fairly recent application in medicine is the use of
elliptical reflectors and ultrasound to break up kidney stones.
Planet
Sun
F
F
Elliptical dome
Planetary motion
Elliptical gears
(a)
(b)
(c)
FIGURE 9 Uses of elliptical forms.
Johannes Kepler (1571–1630), a German astronomer, discovered that planets
move in elliptical orbits, with the sun at a focus, and not in circular orbits as had been
thought before [Fig. 9(a)]. Figure 9(b) shows a pair of elliptical gears with pivot points
at foci. Such gears transfer constant rotational speed to variable rotational speed, and
9-2
Ellipse
625
vice versa. Figure 9(c) shows an elliptical dome. An interesting property of such a
dome is that a sound or light source at one focus will reflect off the dome and pass
through the other focus. One of the chambers in the Capitol Building in Washington,
D.C., has such a dome, and is referred to as a whispering room because a whispered
sound at one focus can be easily heard at the other focus.
Answers to Matched Problems
1. (A)
y
1
Foci: F
(3, 0), F(3, 0)
Major axis length 4
Minor axis length 2
F
F
2
0
2
x
1
y
(B)
18
F
6
Foci: F
(0, 12), F(0, 12)
Major axis length 218 8.49
Minor axis length 26 4.90
6
x
F
18
2. (A)
EXERCISE
x2
y2
1
625 225
(B)
y2
x2
1
100 64
9-2
A
11. 4x2 7y2 28
In Problems 1–6, sketch a graph of each equation, find the
coordinates of the foci, and find the lengths of the major and
minor axes.
In Problems 13–18, find an equation of an ellipse in the
form
x2 y2
1
M, N 0
M N
1.
x2
y2
1
25
4
2.
x2 y2
1
9
4
3.
x2
y2
1
4
25
4.
x2 y2
1
4
9
5. x2 9y2 9
6. 4x2 y2 4
12. 3x2 2y2 24
if the center is at the origin, and:
13. Major axis on y axis
Major axis length 6
Minor axis length 2
B
14. Major axis on x axis
Major axis length 32
Minor axis length 30
In Problems 7–12, sketch a graph of each equation, find the
coordinates of the foci, and find the lengths of the major and
minor axes.
15. Major axis on x axis
Minor axis length 10
Distance of foci from center 4
7. 25x2 9y2 225
9. 2x2 y2 12
8. 16x2 25y2 400
10. 4x2 3y2 24
16. Major axis on y axis
Major axis length 16
Distance of foci from center 7
626
9 Additional Topics in Analytic Geometry
17. Major axis on y axis
Major axis length 24
Distance between foci 2
In Problems 35–38, use a graphing utility to find the coordinates of all points of intersection to two decimal places.
18. Major axis on x axis
Major axis length 4
Distance between foci 50
36. 8x2 35y2 3,600, x2 25y
19. Explain why an equation whose graph is an ellipse does not
define a function.
38. 2x2 7y2 95, 13x2 6y2 63
35. x2 3y2 20, 4x 5y 11
37. 50x2 4y2 1,025, 9x2 2y2 300
20. Consider all ellipses having (0, 1) as the ends of the minor axis. Describe the connection between the elongation
of the ellipse and the distance from a focus to the origin.
APPLICATIONS
39. Engineering. The semielliptical arch in the concrete bridge
in the figure must have a clearance of 12 feet above the water and span a distance of 40 feet. Find the equation of the
ellipse after inserting a coordinate system with the center of
the ellipse at the origin and the major axis on the x axis. The
y axis points up, and the x axis points to the right. How
much clearance above the water is there 5 feet from the
bank?
In Problems 21–24, graph each system of equations in the
same rectangular coordinate system and find the coordinates
of any points of intersection. Find noninteger coordinates to
three decimal places.
Check Problems 21–24 with a graphing utility.*
21. 16x2 25y2 400
2x 5y 10
22. 25x2 16y2 400
5x 8y 20
23. 25x2 16y2 400
25x2 36y 0
24. 16x2 25y2 400
3x2 20y 0
In Problems 25–28, find the first-quadrant points of intersection for each system of equations to three decimal places.
Check Problems 25–28 with a graphing utility.
25. 5x2 2y2 63
2x y 0
26. 3x2 4y2 57
x 2y 0
27. 2x2 3y2 33
x2 8y 0
28. 3x2 2y2 43
x2 12y 0
Elliptical bridge
40. Design. A 4 8 foot elliptical tabletop is to be cut out of a
4 8 foot rectangular sheet of teak plywood (see the figure). To draw the ellipse on the plywood, how far should
the foci be located from each edge and how long a piece of
string must be fastened to each focus to produce the ellipse
(see Figure 1 in the text)? Compute the answer to two decimal places.
In Problems 29–32, determine whether the statement is true
or false. If true, explain why. If false, give a counterexample.
29. The line segment joining the foci of an ellipse has greater
length than the minor axis.
30. There is exactly one ellipse with center (0, 0) and foci
(1, 0).
31. Every line through the center of x2 4y2 16 intersects
the ellipse in exactly two points.
32. Every nonvertical line through a vertex of x2 4y2 16 intersects the ellipse in exactly two points.
String
F
F
C
33. Find an equation of the set of points in a plane, each of
whose distance from (2, 0) is one-half its distance from the
line x 8. Identify the geometric figure.
34. Find an equation of the set of points in a plane, each of
whose distance from (0, 9) is three-fourths its distance from
the line y 16. Identify the geometric figure.
*Please note that use of a graphing utility is not required to complete these exercises. Checking them with a g.u. is optional.
Elliptical table
★
41. Aeronautical Engineering. Of all possible wing shapes, it
has been determined that the one with the least drag along
the trailing edge is an ellipse. The leading edge may be a
straight line, as shown in the figure. One of the most famous planes with this design was the World War II British
Spitfire. The plane in the figure has a wingspan of 48.0 feet.
9-3 Hyperbola
Leading edge
Fuselage
627
and 1.00 foot in front of it. The chord is 1.00 foot shorter
than the major axis.
Trailing edge
Elliptical
wings and tail
(A) If the straight-line leading edge is parallel to the major
axis of the ellipse and is 1.14 feet in front of it, and if
the leading edge is 46.0 feet long (including the width
of the fuselage), find the equation of the ellipse. Let the
x axis lie along the major axis (positive right), and let
the y axis lie along the minor axis (positive forward).
(B) How wide is the wing in the center of the fuselage (assuming the wing passes through the fuselage)?
Compute quantities to 3 significant digits.
★
42. Naval Architecture. Currently, many high-performance
racing sailboats use elliptical keels, rudders, and main sails
for the same reasons stated in Problem 41—less drag along
the trailing edge. In the accompanying figure, the ellipse
containing the keel has a 12.0-foot major axis. The straightline leading edge is parallel to the major axis of the ellipse
SECTION
9-3
Rudder
Keel
(A) Find the equation of the ellipse. Let the y axis lie along
the minor axis of the ellipse, and let the x axis lie along
the major axis, both with positive direction upward.
(B) What is the width of the keel, measured perpendicular
to the major axis, 1 foot up the major axis from the bottom end of the keel?
Compute quantities to 3 significant digits.
Hyperbola
•
•
•
•
Definition of a Hyperbola
Drawing a Hyperbola
Standard Equations and Their Graphs
Applications
As before, we start with a coordinate-free definition of a hyperbola. Using this definition, we show how a hyperbola can be drawn and we derive standard equations for
hyperbolas specially located in a rectangular coordinate system.
• Definition of
a Hyperbola
The following is a coordinate-free definition of a hyperbola: