precalculus warm-up graph the conic. find center, vertices, and foci

Copyright © 2010 Pearson Education, Inc.

HyperbolasHyperbolas

♦ Find equations of hyperbolasFind equations of hyperbolas

♦ Graph hyperbolasGraph hyperbolas

♦ Learn the reflective property of Learn the reflective property of hyperbolashyperbolas

♦ Translate hyperbolasTranslate hyperbolas

9.39.3

Hyperbola

The set of all co-planar points whose difference of the distances from two fixed points (foci) are constant.

Hyperbola

Co-vertices endpoints ofconjugate axis

1)()(

2

2

2

2

b

ky

a

hx

Center: (h, k)

Hyperbola

1)()(

2

2

2

2

b

hx

a

ky

Co-vertices endpoints ofconjugate axis

Hyperbola

c2 = a2 + b2

Slide 7.3 - 14Copyright © 2010 Pearson Education, Inc.

Standard Equation for HyperbolasCentered at (h, k)

The hyperbola with center (h, k), and a horizontal transverse axis satisfies the following equation, where c2 = a2 + b2.

Vertices: (h ± a, k)Foci: (h ± c, k)Asymptotes:

x h 2a2

y k 2

b21

y

b

ax h k

Slide 7.3 - 15Copyright © 2010 Pearson Education, Inc.

Standard Equation for HyperbolasCentered at (h, k)

The hyperbola with center (h, k), and a vertical transverse axis satisfies the following equation, where c2 = a2 + b2.

Vertices: (h, k ± a)Foci: (h, k ± c)Asymptotes:

y k 2a2

x h 2

b21

y

a

bx h k

Slide 7.3 - 16Copyright © 2010 Pearson Education, Inc.

Example 1

Sketch the graph of

Label vertices, foci, and asymptotes.

Solution

Equation is in standard form with a = 2 and b = 3. It has a horizontal transverse axis with vertices (±2, 0) Endpoints of conjugate axis are (0, ±3). Find c.

x2

4

y 2

91.

c2 a2 b2 4 9 13, or c 13 3.61

Slide 7.3 - 17Copyright © 2010 Pearson Education, Inc.

Example 1

Solution continued

foci:

asymptotes:

x2

4

y 2

91

13, 0

y b

ax

y 3

2x

Slide 7.3 - 18Copyright © 2010 Pearson Education, Inc.

Example 2Find the equation of the hyperbola centered at the origin with a vertical transverse axis of length 6 and focus (0, 5). Also find the equations of its asymptotes.

Solution

Since the hyperbola is centered at the origin with a vertical axis, its equation is

y 2

a2

x2

b21

Slide 7.3 - 19Copyright © 2010 Pearson Education, Inc.

Example 2Solution continuedTransverse axis has length 6 = 2a, so a = 3. One focus is (0, 5) so c = 5. Find b.

y 2

9

x2

161

b2 c2 a2

b c2 a2 52 32 4

y

a

bx, or y

3

4x

Standard equation is

asymptotes are

Slide 7.3 - 20Copyright © 2010 Pearson Education, Inc.

Example 5

Graph the hyperbola whose equation is

Label the vertices, foci, and asymptotes.

Solution

Vertical transverse axis. Center: (2, –2)

a2 = 9, b2 = 16 c2 = a2 + b2 = 9 + 16 = 25.

a = 3, b = 4, c = 5

(y 2)2

9

(x 2)2

161.

Slide 7.3 - 21Copyright © 2010 Pearson Education, Inc.

Example 5

Solution continuedCenter: (2, 2) a = 3, b = 4, c = 5

Vertices 3 units above and below center

(2, 1), (2, –5)

Foci 5 units aboveand below center:

(2, 3), (2, –7)

Asymptotes:

y

3

4(x 2) 2

Slide 7.3 - 22Copyright © 2010 Pearson Education, Inc.

Example 6Write 9x2 – 18x – 4y2 – 16y = 43 in the standard form for a hyperbola centered at (h, k). Identify the center, vertices and foci.

Solution:

9x2 18x 4y 2 16y 43

9 x2 2x __ 4 y 2 4y __ 43

9(x2 2x 1) 4(y 2 4y 4) 43 9 16

Slide 7.3 - 23Copyright © 2010 Pearson Education, Inc.

Example 6Solution continued

The center is (1, –2). Because a = 2 and the transverse axis is horizontal, the vertices are (1 ± 2, –2). The foci are (1 ± , –2).

9(x2 2x 1) 4(y 2 4y 4) 43 9 16

9(x 1)2 4(y 2)2 36

(x 1)2

4

(y 2)2

91

13

Graph the following Hyperbola. Find the vertices, foci and asymptotes

149

)5(

16

)1( 22

yx

Center: (-1, 5)a = 4 in x directionb = 7 in y direction

Graph the following Hyperbola. Find the vertices, foci and asymptotes.

149

)5(

16

)1( 22

yx

Center: (-1, 5)a = 4 b = 7

a2 + b2 = c2

42 + 72 = c2

65 = c2

c65

16 + 49 = c2

)5,651( )5,651(

Graph the following Hyperbola. Find the vertices, foci and asymptotes

149

)5(

16

)1( 22

yx

Asymptotes

4

7m

75 ( 1)

4y x

Graph the following Hyperbola. Find the vertices, foci and asymptotes

149

)5(

16

)1( 22

yx

Asymptotes

4

7m

75 ( 1)

4y x

Graph the following Hyperbola. Find the vertices, foci and asymptotes

149

)5(

16

)1( 22

yx

Asymptotes:

Center: (-1, 5)

Vertices: (-5, 5) (3, 5)

Co-Vertices: (-1, 12) (-1, -2)

)5,651( Foci:

7( 1) 54

y x

Homework:pg. 656 1-41 odd

Precalculus Random Conics HWQ

• Find the standard form of the equation of a parabola with vertex

(-2, 1) and directrix at x=1

Graph the following Hyperbola

136

)4(9

36

)2(4 22

yx

4x2 + 16x - 9y2 + 72y - 5 = 874x2 + 16x - 9y2 + 72y = 87 + 5

4(x2 + 4x + 22) - 9(y2 - 8y + (-4)2) = 92 + 16 - 144

4(x + 2)2 - 9(y - 4)2 = -36

19

)2(

4

)4( 22

xy

Graph the following Hyperbola

19

)2(

4

)4( 22

xy

Center: (-2, 4)a = 2 in y directionb = 3 in x direction

Graph the following Hyperbola

Center: (-2, 4)a = 2 b = 3

a2 + b2 = c2

22 + 32 = c2

13 = c2

c13

4 + 9 = c2

)134,2(

)134,2(

19

)2(

4

)4( 22

xy

Graph the following Hyperbola

)134,2(

19

)2(

4

)4( 22

xy

)134,2(

Asymptotes3

2m

))2((3

24 xy

)2(3

24 xy

3

4

3

24 xy

3

16

3

2 xy

Graph the following Hyperbola

)134,2(

19

)2(

4

)4( 22

xy

)134,2(

Asymptotes3

2m

))2((3

24

xy

)2(3

24

xy

3

4

3

24

xy

3

8

3

2

xy

Graph the following Hyperbola

)134,2(

19

)2(

4

)4( 22

xy

)134,2(

Asymptotes

Center: (-2, 4)

Vertices: (-2, 6) (-2, 2)

Co-Vertices: (-5, 4) (1, 4)

Length of Transverse axis: 4

Length of Conjugate axis: 6

)134,2( Foci:

3

16

3

2 xy

3

8

3

2

xy

Slide 7.3 - 37Copyright © 2010 Pearson Education, Inc.

Trajectory of a Comet

One interpretation of an asymptote relates to trajectories of comets as they approach the sun. Comets travel inparabolic, elliptic, or hyperbolic trajectories. If the speed of a comet is too slow, the gravitational pull of the sun will capture the comet in an elliptical orbit.

Slide 7.3 - 38Copyright © 2010 Pearson Education, Inc.

Trajectory of a Comet

If the speed of the comet is too fast, the comet will pass by the sun once in a hyperbolic trajectory; farther from the sun, gravity becomes weaker and the comet will eventually return to a straight-line trajectory that is determined by the asymptote of the hyperbola.

Slide 7.3 - 39Copyright © 2010 Pearson Education, Inc.

Trajectory of a Comet

Finally, if the speed is neither too slow nor too fast, the comet will travel in a parabolic path.

In all three cases, the sun is located at a focus of the conic section.

Slide 7.3 - 40Copyright © 2010 Pearson Education, Inc.

Reflective Property of Hyperbolas

Hyperbolas have an important reflective property. If a hyperbola is rotated about the x-axis, a hyperboloid is formed.

Slide 7.3 - 41Copyright © 2010 Pearson Education, Inc.

Reflective Property of Hyperbolas

Any beam of light that is directed toward focus F1 will be reflected by the hyperboloid toward focus F2.

Homework:pg. Conics Worksheet