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

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


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


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


Example 1

Solution continued

foci:

asymptotes:

x2

4

y 2

91

13, 0

y b

ax

y 3

2x


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


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


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.


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


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


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(


149

)5(

16

)1( 22

yx

Asymptotes

4

7m

75 ( 1)

4y x


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


19

)2(

4

)4( 22

xy

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


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


)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


)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


)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


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.



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.



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.


Reflective Property of Hyperbolas

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


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

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

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