Copyright © 2011 Pearson Education, Inc. Slide 6.2-1.

Copyright © 2011 Pearson Education, Inc. Slide 6.2-1


Chapter 6: Analytic Geometry

6.1 Circles and Parabolas

6.2 Ellipses and Hyperbolas

6.3 Summary of the Conic Sections

6.4 Parametric Equations


6.2 Ellipses and Hyperbolas

• The graph of an ellipse

is not that of a function.• The foci lie on the major

axis – the line from V to

V.• The minor axis – B to B

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points is constant. Each fixed point is called a focus of the ellipse.


6.2 The Equation of an Ellipse

• Let the foci of an ellipse be at the points (c, 0). The sum of the distances from the foci to a point (x, y) on the ellipse is 2a.

So, we rewrite the following equation.

2222

22

)())(()',(

)(),(

ycxycxFPd

ycxFPd

aycxycx 2)()( 2222



1)(

)()(

22

2))((

)(

44)(4

2)(442

)()(44)(

)(2)(

2)()(

22

2

2

2

22222222

224222222

22242222222

2224222

222

222

222222222

2222222

2222

2222

ca

y

a

x

caayacax

caayaxcxa

xcxcaayacaxcaxa

xcxcaaycxa

cxaycxa

cxaycxa

yccxxycxaayccxx

ycxycxaaycx

ycxaycx

aycxycx



• Replacing a2 – c2 with b2 gives the standard equation of an ellipse with the foci on the x-axis.

• Similarly, if the foci were on the y-axis, we get

1

1)(

2

2

2

2

22

2

2

2

by

ax

cay

ax

.12

2

2

2

ay

bx



The ellipse with center at the origin and equation

has vertices (a, 0), endpoints of the minor axis (0, b), and foci (c, 0), where c2 = a2 – b2.

The ellipse with center at the origin and equation

has vertices (0, a), endpoints of the minor axis (b, 0), and foci (0, c), where c2 = a2 – b2.

2 2

2 21 0

x ya b

a b

2 2

2 21 0

x ya b

b a


6.2 Graphing an Ellipse Centered at the Origin

Example Graph

Solution Divide both sides by 36.

This ellipse, centered at the origin, has x-intercepts 3 and –3, and y-intercepts 2 and –2. The domain is [–3, 3]. The range is [–2, 2].

.3694 22 yx

149

22

yx


6.2 Finding Foci of an Ellipse

Example Find the coordinates of the foci of the equation

Solution From the previous example, the equation of the ellipse in standard form isSince 9 > 4, a2 = 9 and b2 = 4.

The major axis is along the x-axis, so the foci have coordinates

.3694 22 yx

.149

22

yx

5

5492

222

c

c

bac

).0,5(and)0,5(


6.2 Finding the Equation of an Ellipse

Example Find the equation of the ellipse having center at the origin, foci at (0, ±3), and major axis oflength 8 units. Give the domain and range.

Solution 2a = 8, so a = 4.

Foci lie on the y-axis, so the larger intercept, a, is usedto find the denominator for y2. The standard form is

with domain and range [–4, 4].

7

342

222

222

b

b

cba

,1716

22

xy ]7,7[


6.2 Ellipse Centered at (h, k)

An ellipse centered at (h, k) and either a horizontal or vertical major axis satisfies one of the following equations, where a > b > 0, and with c > 0:

2 2

2 2

( ) ( )1

x h y k

a b

2 2

2 2

( ) ( )1

x h y k

b a

2 2 2c a b



Example Graph

Analytic Solution Center at (2, –1). Since a > b, a = 4 is associated with the y2 term, so the vertices are on the vertical line through (2, –1).

.116

)1(9

)2( 22

yx



Graphical SolutionSolving for y in the equation yields

The + sign indicates the upper half of the ellipse, while the – sign yields the bottom half.

.9

)2(141

2 xy


6.2 Finding the Standard Form of an Ellipse

Example Write the equation in standard form.

Solution

Center (2, –3); Vertices (2±3, –3) = (5, –3), (–1, –3)

2 24 16 9 54 61 0x x y y

2 2

2 2

2 2

2 2

4 16 9 54 61 04 4 4 9 6 9 61 4 4 9 9

4 2 9 3 36

2 31

9 4

x x y yx x y y

x y

x y


6.2 Hyperbolas

• If the center is at the origin, the

foci are at (±c, 0).

• The midpoint of the line segment

F F is the center of the hyperbola.

• The vertices are at (±a, 0).

• The line segment VV is called

the transverse axis.

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points is constant. The two fixed points are called the foci of the hyperbola.


6.2 Standard Forms of Equations for Hyperbolas

The hyperbola with center at the origin and equation

has vertices (±a, 0), asymptotes y = ±b/ax, and foci (±c, 0), where c2 = a2 + b2.

The hyperbola with center at the origin and equation

has vertices (0, ±a), asymptotes y = ±a/bx, and foci (0, ±c), where c2 = a2 + b2.

12

2

2

2

by

ax

12

2

2

2

bx

ay


6.2 Standard Forms of Equations for Hyperbolas

Solving for y in the first equation gives

If |x| is large, the difference approaches x2. Thus, the hyperbola has asymptotes

.22 axyab

22 ax

.xaby


6.2 Using Asymptotes to Graph a Hyperbola

Example Sketch the asymptotes and graph the hyperbola

Solution a = 5 and b = 7

Choosing x = 5 (or –5) gives y = ±7. These four points: (5, 7), (–5, 7), (5, –7), and (–5, –7), are the corners of the fundamental rectangle shown.The x-intercepts are ±5.

.14925

22

yx

xxab

y57


6.2 Graphing a Hyperbola with the Graphing Calculator

Example Graph

Solution Solve the given equation for y.

.9425 22 xy

.4951

and ,4951

Let

22

21

xy

xy

2

2

22

22

4951

4954925

9425

xy

xyxy

xy


6.2 Graphing a Hyperbola Translated from the Origin

Example Graph

Solution This hyperbola has the same graph as

except that it is centered at (–3, –2).

.14

)3(9

)2( 22

xy

,149

22

xy


6.2 Finding the Standard Form for a Hyperbola

Example Write the equation in standard form.

Solution

Center (1, –2); Vertices (1±2, –2) = (3, –2), (–1, –2)

2 29 18 4 16 43x x y y

2 2

2 2

2 2

2 2

9 18 4 16 439 2 1 4 4 4 43 9 1 4 4

9 1 4 2 36

1 21

4 9

x x y yx x y y

x y

x y

Copyright © 2011 Pearson Education, Inc. Slide 6.2-1.

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