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C. Characteristics of the Conic Sections

C. Characteristics of the Conic Sections

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Section 8.1 A Brief Introduction to Analytical Geometry



EXAMPLE 3







711



Finding an Equation for All Points That Form a Certain Parabola

With Example 2 as a pattern, use the analytic definition to find a formula

(equation) for the set of all points that form the parabola.



Solution



Y1 ϭ







1 2

X , Y2 ϭ Ϫ2

8



Use the ordered pair (x, y) to represent an arbitrary point on the parabola. Since

any point on the line y ϭ Ϫ2 has coordinates 1x, Ϫ22 , we set the distance from

1x, Ϫ22 to (x, y) equal to the distance from (0, 2) to (x, y). The result is



9



Ϫ12



12



Ϫ6



21x Ϫ x2 2 ϩ 3y Ϫ 1Ϫ22 4 2 ϭ 21x Ϫ 02 2 ϩ 1y Ϫ 22 2 distances are equal

simplify

21y ϩ 22 2 ϭ 2x2 ϩ 1y Ϫ 22 2

power property

1y ϩ 22 2 ϭ x2 ϩ 1y Ϫ 22 2

2

2

2

expand binomials

y ϩ 4y ϩ 4 ϭ x ϩ y Ϫ 4y ϩ 4

simplify

8y ϭ x2

1

result

y ϭ x2

8

All points satisfying these conditions are on the parabola defined by y ϭ 18x2.

See the figure.

Now try Exercises 27 and 28







At this point, it seems reasonable to ask what happens when the distance from the

focus to (x, y) is less than the distance from the directrix to (x, y). For example, what if the

distance is only two-thirds as long? As you might guess, the result is one of the other conic

sections, in this case an ellipse. If the distance from the focus to a point (x, y) is greater

than the distance from the directrix to (x, y), one branch of a hyperbola is formed. While

we will defer a development of their general equations until later in the chapter, the following diagrams serve to illustrate this relationship for the ellipse, and show why we refer

to the conic sections as a family of curves. In Figure 8.8, the line segment from the focus

to each point on the graph (shown in blue), is exactly two-thirds the length of the line segment from the directrix to the same point (shown in red). Note the graph of these points

forms the right half of an ellipse. In Figure 8.9, the lines and points forming the first half

are removed to more clearly show the remaining points that form the complete graph.

Figure 8.8



EXAMPLE 4







Figure 8.9



Finding an Equation for All Points That Form a Certain Ellipse

Suppose we arbitrarily select the point (1, 0) as a focus and the (vertical) line x ϭ 4 as

the directrix. Use these to find an equation for the set of all points where the distance

from the focus to a point (x, y) is 12 the distance from the directrix to (x, y).



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CHAPTER 8 Analytic Geometry and the Conic Sections



Solution







Since any point on the line x ϭ 4 has coordinates (4, y), we have:

1

Distance from 11, 02 to 1x, y2 ϭ 3 distance from 14, y2 to 1x, y2 4 in words

2

1

21x Ϫ 12 2 ϩ 3y Ϫ 102 4 2 ϭ 21x Ϫ 42 2 ϩ 1y Ϫ y2 2 resulting equation

2

1

21x Ϫ 12 2 ϩ y2 ϭ 21x Ϫ 42 2

simplify

2

1

1x Ϫ 12 2 ϩ y2 ϭ 1x Ϫ 42 2

power property

4

1

x2 Ϫ 2x ϩ 1 ϩ y2 ϭ 1x2 Ϫ 8x ϩ 162

expand binomials

4

1

x2 Ϫ 2x ϩ 1 ϩ y2 ϭ x2 Ϫ 2x ϩ 4

distribute

4

3 2

1

3

x ϩ y2 ϭ 3

simplify: 1x 2 Ϫ x 2 ϭ x 2

4

4

4

3x2 ϩ 4y2 ϭ 12

polynomial form

All points satisfying these conditions are on the ellipse defined by 3x2 ϩ 4y2 ϭ 12.

Now try Exercises 29 and 30



Figure 8.10

f1



f2



C. You’ve just seen how

we can use the defining

characteristics of a conic

section to find its equation







Actually, any given ellipse has two foci (see Figure 8.10) and the equation from

Example 4 could also have been developed using the left focus (with the directrix also

on the left). This symmetrical relationship leads us to an alternative definition for the

ellipse, which we will explore further in Section 8.2:

For foci f1 and f2, an ellipse is the set of all points

Figure 8.11

(x, y) where the sum of the distances from f1 to (x, y)

and f2 to (x, y) is constant.

d1

d2

See Figure 8.11 and Exercises 31 and 32. Both the

f1

focus/directrix definition and the two foci definition have

d3

f2

d4

merit, and simply tend to call out different characteris- (x, y)

tics and applications of the ellipse. The hyperbola also

d1 ϩ d2 ϭ d3 ϩ d4

has a focus/directrix definition and a two foci definition.

See Exercises 33 and 34.



8.1 EXERCISES





CONCEPTS AND VOCABULARY



Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.



1. Analytical geometry is a study of

the tools of

.



using



2. The distance formula is d ϭ

the midpoint formula is M ϭ



;

.



3. The distance between a point and a line always

refers to the

distance.



4. The conic sections are formed by the intersection

of a

and a

.



5. If a plane intersects a cone at its vertex, the result is

a

, a line, or a pair of

lines.



6. A circle is defined relative to an equal distance

between two

. A parabola is defined relative

to an equal distance between a

and a

.



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DEVELOPING YOUR SKILLS



The three points given form a right triangle. Find the

midpoint of the hypotenuse and verify that the midpoint

is an equal distance from all three vertices.



7. P1 ϭ 1Ϫ5, 22

P2 ϭ 11, 22

P3 ϭ 1Ϫ5, Ϫ62

9. P1 ϭ 1Ϫ2, 12

P2 ϭ 16, Ϫ52

P3 ϭ 12, Ϫ72



11. P1 ϭ 110, Ϫ212

P2 ϭ 1Ϫ6, Ϫ92

P3 ϭ 13, 32



8. P1 ϭ 13, 22

P2 ϭ 13, 142

P3 ϭ 18, 22



16. Find an equation of the circle that circumscribes

the triangle in Exercise 10.

17. Find an equation of the circle that circumscribes

the triangle in Exercise 11.

18. Find an equation of the circle that circumscribes

the triangle in Exercise 12.



10. P1 ϭ 10, Ϫ52

P2 ϭ 1Ϫ6, 42

P3 ϭ 16, Ϫ12



19. Of the following six points, four are an equal

distance from the point A(2, 3) and two are not.

(a) Identify which four, and (b) find any two

additional points that are this same (nonvertical,

nonhorizontal) distance from (2, 3):

B(7, 15)

D(9, 14)

C1Ϫ10, 82

E1Ϫ3, Ϫ92



12. P1 ϭ 16, Ϫ62

P2 ϭ 1Ϫ12, 182

P3 ϭ 120, 422



F15, 4 ϩ 3 1102



13. Find an equation of the circle that circumscribes

the triangle in Exercise 7.



G12 Ϫ 2 130, 102



20. Of the following six points, four are an equal

distance from the point P1Ϫ1, 42 and two are not.

(a) Identify which four, and (b) find any two

additional points that are the same (nonvertical,

nonhorizontal) distance from (Ϫ1, 4).

Q1Ϫ9, 102 R(5, 12) S1Ϫ7, 112 T14, 4 ϩ 5 132



14. Find an equation of the circle that circumscribes

the triangle in Exercise 8.

15. Find an equation of the circle that circumscribes

the triangle in Exercise 9.







U1Ϫ1 ϩ 416, 62



V1Ϫ7, 4 ϩ 1512



WORKING WITH FORMULAS



The Perpendicular Distance from a Point to a Line: d ‫؍‬



ͦAx1 ؉ By1 ؉ Cͦ



(x1, y1)



. The perpendicular



2A2 ؉ B2

distance from a point (x1, y1) to a given line can be found using the formula shown, where

Ax ؉ By ؉ C ‫ ؍‬0 is the equation of the line in standard form (A, B, and C are integers).



21. Use the formula to verify that P1Ϫ6, 22 and Q(6, 4)

are an equal distance from the line y ϭ Ϫ12x ϩ 3.







713



Section 8.1 A Brief Introduction to Analytical Geometry



d



Ax ϩ By ϩ C ϭ 0



22. Find the value(s) for y that ensure

(1, y) is this same distance from

y ϭ Ϫ12x ϩ 3.



APPLICATIONS



23. Of the following four points, three are an equal

distance from the point A(0, 1) and the line

y ϭ Ϫ1. (a) Identify which three, and (b) find any

two additional points that satisfy these conditions.

B1Ϫ6, 92

C14, 42

D1Ϫ2 12, 62

E14 12, 82

24. Of the following four points, three are an equal

distance from the point P(2, 4) and the line

y ϭ Ϫ4. (a) Identify which three, and (b) find any

two additional points that satisfy these conditions.

Q1Ϫ10, 92

R12 ϩ 4 12, 32

S110, 42

T12 Ϫ 4 15, 52



25. Consider the fixed point 10, Ϫ42 and the fixed line

y ϭ 4. Verify that the distance from each point

given to 10, Ϫ42 , is equal to the distance from the

point to the line y ϭ 4.

25

C1412, Ϫ22

A14, Ϫ12

B a10, Ϫ b

4

D18 15, Ϫ202



26. Consider the fixed point 10, Ϫ22 and the fixed line

y ϭ 2. Verify that the distance from each point

given to 10, Ϫ22 , is equal to the distance from the

point to the line y ϭ 2.

9

R1415, Ϫ102

P112, Ϫ182

Qa6, Ϫ b

2

S14 16, Ϫ122



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27. The points from Exercise 25 are on the graph of a

parabola. Find an equation of the parabola.

28. The points from Exercise 26 are on the graph of a

parabola. Find an equation of the parabola.



29. Using 10, Ϫ22 as the focus and the horizontal line

y ϭ Ϫ8 as the directrix, find an equation for the set

of all points (x, y) where the distance from the

focus to (x, y) is one-half the distance from the

directrix to (x, y).

30. Using (4, 0) as the focus and the vertical line x ϭ 9 as

the directrix, find an equation for the set of all points

(x, y) where the distance from the focus to (x, y) is

two-thirds the distance from the directrix to (x, y).

31. From Exercise 29, verify the points 1Ϫ3, 22 and

1 112, 02 are on the ellipse defined

by 4x2 ϩ 3y2 ϭ 48. Then verify that

d1 ϩ d2 ϭ d3 ϩ d4.

y

5



f2 (0, 2)



(Ϫ3, 2)

d1

Ϫ5



d2

(0, Ϫ2)



d3



( 12, 0)

5 x



f1



d4



Ϫ5



Exercise 32

y

5



΂4, j΃

d1

(Ϫ4, 0)

Ϫ6 f1



f2



(4, 0) 6 x



d3

(Ϫ3, Ϫ 15)



d2



d4

Ϫ5



33. From the focus/directrix

definition of a hyperbola: If the distance from the

focus to a point (x, y) is greater than the distance

from the directrix to (x, y), one branch of a

hyperbola is formed. Using (2, 0) as the focus and

the vertical line x ϭ 12 as the directrix, find an

equation for the set of all points (x, y) where the

distance from the focus to (x, y), is twice the

distance from the directrix to (x, y).



5 x



EXTENDING THE CONCEPT



35. Properties of a circle: A

theorem from elementary

geometry states: If a

radius is perpendicular

to a chord, it bisects the

chord. Verify this is true

for the circle, radii, and

chords shown.







32. From Exercise 30, verify

the points 14, 10

3 2 and

1Ϫ3, Ϫ1152 are on the

ellipse defined by

5x2 ϩ 9y2 ϭ 180.

Then verify that

d1 ϩ d2 ϭ d3 ϩ d4.



y

34. From the two foci definition

5

of a hyperbola: For foci f1

(2, 3)

and f2, a hyperbola is the set

(Ϫ2, 0) d1

d2

f2

of all points (x, y) where the

(2, 0)

f1

Ϫ5

difference of the distances

d4

from f1 to (x, y) and f2 to (x, y)

d

is constant. Verify the points (Ϫ3, Ϫ2 6) Ϫ53

(2, 3) and 1Ϫ3, Ϫ2162 are

on the graph of the hyperbola from Exercise 33.

Then verify d1 Ϫ d2 ϭ d3 Ϫ d4.



Exercise 31







8–8



CHAPTER 8 Analytic Geometry and the Conic Sections



y

5



(Ϫ3, 4)

Q

(Ϫ4, 2)

T



C



P

Ϫ5



5 x



S

(2, Ϫ4) U

Ϫ5



(4, Ϫ3)



R



36. Verify that points C1Ϫ2, 32 and D12 12, 162 are

points on the ellipse with foci at A1Ϫ2, 02 and B(2, 0),

by verifying d1AC2 ϩ d1BC2 ϭ d1AD2 ϩ d1BD2.

The expression that results has the form

1U ϩ V ϩ 1U Ϫ V, which prior to the common

use of technology, had to be simplified using the

formula 1U ϩ V ϩ 1U Ϫ V ϭ 2a ϩ 1b,

where a ϭ 2U and b ϭ 41U2 Ϫ V2 2 . Use this

relationship to simplify the equation above.



MAINTAINING YOUR SKILLS



37. (5.6) $5000 is deposited at 4% compounded

continuously. How many years will it take for the

account to exceed $8000?



39. (4.3) Use the rational zeroes theorem and other

tools to factor f (x) and sketch its graph:

f 1x2 ϭ x4 Ϫ 3x3 Ϫ 3x2 ϩ 11x Ϫ 6.



38. (5.5) Solve for x in both exact and approximate form:

10

a. 5 ϭ

b. 345 ϭ 5e0.4x ϩ 75

1 ϩ 9eϪ0.5x



40. (4.4) Sketch a complete graph of h1x2 ϭ



x2 Ϫ 9

.

x2 Ϫ 4

Clearly label all intercepts and asymptotes.



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8.2



The Circle and the Ellipse



LEARNING OBJECTIVES

In Section 8.2 you will see how

we can:



A. Use the characteristics of

a circle and its graph to

understand the equation

of an ellipse

B. Use the equation of an

ellipse to graph central

and noncentral ellipses

C. Locate the foci of an

ellipse and use the foci

and other features to

write the equation

D. Solve applications

involving the foci



EXAMPLE 1



In Section 8.1, we introduced the equation of an ellipse using analytical geometry and

the focus-directrix definition. Here we’ll take a different approach, and use the equation of a circle to demonstrate that a circle is simply a special ellipse. In doing so, we’ll

establish a relationship between the foci and vertices of the ellipse, that enables us to

apply these characteristics in context.



A. The Equation and Graph of a Circle

Recall that the equation of a circle with radius r and center at (h, k) is

1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r2.



As in Section 1.1, the standard form can be used to construct the equation of the

circle given the center and radius as in Example 1, or to graph the circle as in Example 2.







Determining the Equation of a Circle Given Its Center and Radius

Find an equation of the circle with radius 5 and center at (2, Ϫ1), then graph the

relation on a calculator.



Solution







With a center of (2, Ϫ1), we have h ϭ 2, k ϭ Ϫ1, and r ϭ 5. Making the

corresponding substitutions into the standard form we obtain

1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r2

1x Ϫ 22 2 ϩ 3y Ϫ 1Ϫ12 4 2 ϭ 52

1x Ϫ 22 2 ϩ 1y ϩ 12 2 ϭ 25



standard form

substitute 2 for h, Ϫ1 for k, and 5 for r

simplify



The equation of this circle is 1x Ϫ 22 ϩ 1y ϩ 12 2 ϭ 25.

Recall from Section 1.1 that circles (and other relations) can be graphed by

solving for y, then graphing the upper and lower halves of the circle.

2



1x Ϫ 22 2 ϩ 1y ϩ 12 2 ϭ 25

1y ϩ 12 2 ϭ 25 Ϫ 1x Ϫ 22 2

y ϩ 1 ϭ Ϯ 225 Ϫ 1x Ϫ 22 2

y ϭ Ϯ 225 Ϫ 1x Ϫ 22 2 Ϫ 1



original equation

isolate term containing y

take square roots

subtract 1



Y1 ϭ ϩ 225 Ϫ 1X Ϫ 22 Ϫ 1, Y2 ϭ Ϫ 225 Ϫ 1X Ϫ 22 2 Ϫ 1

2



The graph is shown in the figure using a

square window. Note the point (5, 3)

satisfies the original equation and is a

point on the graph and that 15, Ϫ52,

1Ϫ1, Ϫ52 , and 1Ϫ1, 32 must also be on the

graph due to symmetry.



6.2



Ϫ9.4



9.4



Ϫ6.2



Now try Exercises 7 through 12







If the equation is given in polynomial form, recall that we first complete the square

in x and y to identify the center and radius.

8–9



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CHAPTER 8 Analytic Geometry and the Conic Sections



EXAMPLE 2







Completing the Square to Graph a Circle

Find the center and radius of the circle whose equation is given, then sketch its

graph: x2 ϩ y2 Ϫ 6x ϩ 4y Ϫ 3 ϭ 0.



Solution







Begin by completing the square in both x and y.



1x2 Ϫ 6x ϩ __ 2 ϩ 1y2 ϩ 4y ϩ __ 2 ϭ 3

1x2 Ϫ 6x ϩ 92 ϩ 1y2 ϩ 4y ϩ 42 ϭ 3 ϩ 9 ϩ 4

adds 9 to left side



adds 4 to left side



1x Ϫ 32 2 ϩ 1y ϩ 22 2 ϭ



group x- and y-terms; add 3



complete the square

add 9 ϩ 4 to right side

16

factor and simplify



The center is at (3, Ϫ2), with radius r ϭ 116 ϭ 4.

y

3



(3, 2)



Circle

Center at (3, Ϫ2)



Ϫ2



8



rϭ4

(3, Ϫ2)



(Ϫ1, Ϫ2)



Ϫ7



Radius: r ϭ 4



x



Diameter: 2r ϭ 8



(7, Ϫ2)



Endpoints of horizontal diameter

(Ϫ1, Ϫ2) and (7, Ϫ2)

Endpoints of vertical diameter

(3, 2) and (3, Ϫ6)



(3, Ϫ6)



Now try Exercises 13 through 18







The equation of a circle in standard form provides a useful link to some of the

other conic sections, and is obtained by setting the equation equal to 1. In the case of

a circle, this means we simply divide by r2.

1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r2



1x Ϫ h2

r

A. You’ve just seen how

we can use the characteristics

of a circle and its graph to

understand the equation of an

ellipse



2



2



ϩ



1y Ϫ k2

r2



standard form



2



ϭ1



divide by r 2



In this form, the value of r in each denominator gives the horizontal and vertical

distances, respectively, from the center to the graph. This is not so important in the case

of a circle, since this distance is the same in any direction. But for other conics, these

horizontal and vertical distances are not the same, making the new form a valuable tool

for graphing. To distinguish the horizontal from the vertical distance, r2 is replaced by

a2 in the “x-term” (horizontal distance), and by b2 in the “y-term” (vertical distance).

This distinction leads us directly into our study of the ellipse.



B. The Equation of an Ellipse

It then seems reasonable to ask, “What happens to the graph when a b?” To answer,

1x Ϫ 32 2

1y ϩ 22 2

consider the equation from Example 2. We have

ϩ

ϭ 1 (after

42

42

1x Ϫ 32 2

1y ϩ 22 2

dividing by 16), which we now compare to

ϩ

ϭ 1, where a ϭ 4

42

32

and b ϭ 3. The center of the graph is still at (3, Ϫ2), since h ϭ 3 and k ϭ Ϫ2 remain

unchanged. Substituting y ϭ Ϫ2 to find additional points, eliminates the y-term and

gives two values for x:



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