4 Edition: College Algebra & Trigonometry

College Algebra & Trigonometry 4th EDITION and Precalculus 10TH EDITION
6.4 - 1
10.4
Summary of the Conic Sections

Characteristics Identifying Conic Sections Geometric Definition of Conic Sections
6.4 - 2
Characteristics
The graphs of parabolas, circles, ellipses, and hyperbolas are called conic sections since each graph can be obtained by cutting a cone with a plane, as suggested by Figure 1 at the beginning of the chapter. All conic sections of the types presented in this chapter have equations of the general form
Ax 2 Cy 2 Dx Ey F 0,
where either A or C must be nonzero.
6.4 - 3
Conic Section Parabola Circle
Characteristic Either A = 0, or C = 0, but not both. A=C0
Example x2 y 4 = 0 x + y2 4y = 0 x2 + y2 16 = 0
Ellipse Hyperbola
A C, AC > 0 AC < 0
25x2 + 16y2 400 = 0 x2 y2 1 = 0
6.4 - 4
Equation
(x h)2 = 4p(y k) or y k = a(x h)2
Graph
Opens up if p > 0 (or a > 0), down if p < 0 (or a < 0). Vertex is (h, k). Axis of symmetry is x = h. x2 term y is not squared.
6.4 - 5
Description
Identification
Equation
(y k)2 = 4p(x h) or x h = a(y k)2
Graph
Opens to the right if p > 0 (or a > 0), to the left if p < 0 (or a < 0). Vertex is (h, k). Axis of symmetry is y = k. y2 term x is not squared.
6.4 - 6
Description
Identification
Equation
(x h)2 + (y k)2 = r 2
Graph
Description
Center is (h, k), radius is r.
Identification
x2 and y2 terms have the same positive coefficient.

6.4 - 7
Equation
x2 y 2 2 1 (a b ) 2 a b
Graph
Description
x-intercepts are a and a. y-intercepts are b and b. Horizontal major axis, length = 2a. x2 and y2 terms have different positive coefficients.
6.4 - 8
Identification
Equation
x2 y 2 2 1 (a b ) 2 b a
Graph
Description
x-intercepts are b and b. y-intercepts are a and a. Vertical major axis, length = 2a. x2 and y2 terms have different positive coefficients.
6.4 - 9
Identification
Equation
x2 y 2 2 1 2 a b
Graph
Description
x-intercepts are a and a. Asymptotes are found from (a, b), (a, b),( a, b), and ( a, b). x2 term has a positive coefficient. y2 term has a negative coefficient.
6.4 - 10
Identification
Equation
y 2 x2 2 1 2 a b
Graph
Description
y-intercepts are a and a. Asymptotes are found from (b, a), (b, a),( b, a), and ( b, a). y2 term has a positive coefficient. x2 term has a negative coefficient.
6.4 - 11
Identification
Example 1
DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS
Determine the type of conic section represented by each equation, and graph it.
a. x 2 25 5 y 2 Solution
Divide each term by 25.
x 5 y 25
2 2
Subtract 5y2.
x2 y 2 1 25 5
Divide by 25.
6.4 - 12
Example 1
The equation represents a hyperbola centered at the origin, with asymptotes
x2 y 2 0, or 25 5
5 y x. 5
Remember both the positive and negative square roots.
6.4 - 13
Example 1
The x-intercepts are 5; the graph is shown here.
6.4 - 14
Example 1
2 2
b. x 8 x y 10 y 41 Solution
2
x 2 8 x y 2 10 y 41
2
( x 8 x 16 16) ( y 10y 25 25) 41

Complete the square on both x and y.
( x 2 8 x 16) 16 ( y 2 10 y 25) 25 41
Regroup terms.
( x 4) ( y 5) 41 16 25
2 2
Factor; add 16 and 25.
( x 4)2 ( y 5)2 0
6.4 - 15
Example 1
2 2
b. x 8 x y 10 y 41 Solution
The resulting equation is that of a circle with radius 0; that is, the point (4, 5). If we had obtained a negative number on the right (instead of 0), the equation would have no solution at all, and there would be no graph.
6.4 - 16
Example 1
2 2 4 x 16 x 9 y 54 y 61 c.
Solution The coefficients of the x2 -and y2 -terms are unequal and both positive, so the equation might represent an ellipse but not a circle. (It might also represent a single point or no points at all.)
6.4 - 17
Example 1
2 2 4 x 16 x 9 y 54 y 61 c.
Solution 2 4( x 4 x
2
) 9( y 6 y
2
) 61
Factor out 4; factor out 9.
4( x 4 x 4 4) 9( y 6 y 9 9) 61
2
Complete the square.
4( x 4 x 4) 16 9( y 6 y 9) 81 61
2 2
Multiply. 4( 4) = 16
Distributive property.
6.4 - 18
Example 1
2 2 4 x 16 x 9 y 54 y 61 c.
Solution
4( x 2) 9( y 3) 36
2 2
Factor; add 97.
( x 2) ( y 3) 1 9 4
2 2
Divide by 36.
This equation represents an ellipse having center (2, 3) and graph as shown here.
6.4 - 19
Example 1
2 x d. 6 x 8 y 7 0
Solution Since only one variable is squared (x, and not y), the equation represents a parabola. Get the term with y (the variable that is not squared) alone on one side.
6.4 - 20
Example 1
2 x d. 6 x 8 y 7 0
Solution
8y x 6 x 7
2
Isolate the y-term.
8 y ( x 2 6 x
2
)7
Regroup terms: factor out 1.
8 y ( x 6 x 9 9) 7
6.4 - 21
Example 1
2 x d. 6 x 8 y 7 0
Solution
8 y ( x 6 x 9) 9 7
2
Distributive property; ( 9) = + 9
8 y ( x 3) 16
2
Factor; add.
1 2 y ( x 3) 2 8 1 y 2 ( x 3)2 8
Multiply by .
Subtract 2.
6.4 - 22
Example 1
2 x d. 6 x 8 y 7 0
Solution
The parabola has vertex (3, 2) and opens down, as shown in the graph here. An equivalent form for this parabola is
( x 3) 8( y 2).
2
6.4 - 23
Example 2
DETERMINING THE TYPE OF CONIC SECTION FROM ITS EQUATION
Identify the graph of 4 y 2 16 y 9 x 2 18 x 43.
Solution
4( y 4 y
2
) 9( x 2 x
2
) 43
Factor out 4; factor out 9.
4( y 2 4 y 4 4) 9( x 2 2 x 1 1) 43
6.4 - 24
Example 2
DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION
Solution 2 2 4( y 4 y 4) 16 9( x 2 x 1) 9 43
Distributive property.
4( y 2)2 9( x 1)2 36
Factor; add 16 and subtract 9.
6.4 - 25
Example 2
Solution
4( y 2)2 9( x 1)2 36
Because of the 36, we might think that this equation does not have a graph. However, the minus sign in the middle on the left shows that the graph is that of a hyperbola.
6.4 - 26
Example 2
Solution 2 2 ( x 1) ( y 2) 1 4 9
Divide by 36; rearrange terms.
Be careful here.
This hyperbola has center (1, 2). The graph is shown here.
6.4 - 27
Geometric Definition of Conic Sections

In Section 6.1, a parabola was defined as the set of points in a plane equidistant from a fixed point (focus) and a fixed line (directrix). A parabola has eccentricity 1. This definition can be generalized to apply to the ellipse and the hyperbola.
6.4 - 28

This figure shows an ellipse with a = 4, c = 2, and e = . The line x = 8 is shown also. For any point P on the ellipse,
1 [distance of P from the focus] [distance of P from the line]. 2
6.4 - 29

This figure shows a hyperbola with a = 2, c = 4, and e = 2, along with the line x = 1. For any point P on the hyperbola.
[distance of P from the focus] 2[distance of P from the line].
The following geometric definition applies to all conic sections except circles, which have e = 0.
6.4 - 30
Geometric Definition of a Conic Section

Given a fixed point F (focus), a fixed line L (directrix), and a positive number e, the set of all points P in the plane such that
[distance of P from F ] e [distance of P from L]
is a conic section of eccentricity e. The conic section is a parabola when e = 1, an ellipse when 0 < e < 1, and a hyperbola when e > 1.
6.4 - 31

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College Algebra & Trigonometry 4th EDITION and Precalculus 10TH EDITION

Summary of the Conic Sections

Conic Section Parabola Circle

Characteristic Either A = 0, or C = 0, but not both. A=C0

25x2 + 16y2 400 = 0 x2 y2 1 = 0

(x h)2 = 4p(y k) or y k = a(x h)2

(y k)2 = 4p(x h) or x h = a(y k)2

Center is (h, k), radius is r.

x2 and y2 terms have the same positive coefficient.

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

Remember both the positive and negative square roots.

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

( x 8 x 16 16) ( y 10y 25 25) 41

Factor; add 16 and 25.

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

Factor out 4; factor out 9.

Complete the square.

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

Factor; add 97.

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

Isolate the y-term.

Regroup terms: factor out 1.

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS

DETERMINING THE TYPE OF CONIC SECTION FROM ITS EQUATION

Identify the graph of 4 y 2 16 y 9 x 2 18 x 43.

Factor out 4; factor out 9.

DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION

Identify the graph of 4 y 2 16 y 9 x 2 18 x 43.

Factor; add 16 and subtract 9.

DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION

Identify the graph of 4 y 2 16 y 9 x 2 18 x 43.

DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION

Identify the graph of 4 y 2 16 y 9 x 2 18 x 43.

Geometric Definition of Conic Sections

Geometric Definition of Conic Sections

Geometric Definition of Conic Sections

Geometric Definition of a Conic Section

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