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Presents:

Hyperbola

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3/9/2014 10:20 AM

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SPECIFIC OBJECTIVES:

At the end of the lesson, the student is expected to be able to:

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• give the properties of hyperbola.

• write the standard and general equation of a hyperbola.

• sketch the graph of hyperbola accurately.

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FroydWess - Online Notes

3/9/2014 10:20 AM

A hyperbola is created from the intersection of a plane with a double cone.

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A basketball court where both the keys

and three point lines, are hyperbolas.

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Where are the Hyperbolas?

• A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.

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3/9/2014 10:20 AM

A hyperbola is a set of all such that the difference of the distances from two fixed points is constant.

When you subtract the small line from the long line for each ordered pair the remaining value is the same.

Hyperbolas can be symmetrical around the x-axis or the y-axis. The one on the right is symmetrical around the x-axis.

FroydWess - Online Notes

3/9/2014 10:20 AM

A hyperbola is a set of points in a plane the difference of whose distances from two fixed points, called foci, is a constant.

F₁

F₂

d₁

d₂

P

For any point P that is on the hyperbola, d₂ – d₁ is always the same.

In this example, the origin is the center of the hyperbola. It is midway between the foci.

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F

F

V

V

C

A line through the foci intersects the hyperbola at two points, called the vertices.

The segment connecting the vertices is called the transverse axis of the hyperbola.

The center of the hyperbola is located at the midpoint of the transverse axis.

As x and y get larger the branches of the hyperbola approach a pair of intersecting lines called the asymptotes of the hyperbola. These asymptotes pass through the center of the hyperbola.

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3/9/2014 10:20 AM

F

F

V

V

C

The figure at the left is an example of a hyperbola whose branches open up and down instead of right and left.

Since the transverse axis is vertical, this type of hyperbola is often referred to as a vertical hyperbola.

When the transverse axis is horizontal, the hyperbola is referred to as a horizontal hyperbola.

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3/9/2014 10:20 AM

PARTS OF A HYPERBOLA

center

foci

foci

conjugate axis

vertices

vertices

The black dashes lines are asymptotes for the graphs.

transverse axis

FroydWess - Online Notes

3/9/2014 10:20 AM

General Rules

• x and y are both squared
• Equation always equals(=) 1
• Equation is always minus(-)
• a² is always the first denominator
• c² = a² + b²
• c is the distance from the center to each foci on the major axis
• a is the distance from the center to each vertex on the major axis

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Hyperbola

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3/9/2014 10:20 AM

General Rules

• b is the distance from the center to each midpoint of the rectangle used to draw the asymptotes. This distance runs perpendicular to the distance (a).
• Major axis has a length of 2a
• Eccentricity(e): e = c/a (The closer e gets to 1, the closer it is to being circular
• If x² is first then the hyperbola is horizontal
• If y² is first then the hyperbola is vertical.

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Hyperbola

FroydWess - Online Notes

3/9/2014 10:20 AM

General Rules

• The center is in the middle of the 2 vertices and the 2 foci.
• The vertices and the covertices are used to draw the rectangles that form the asymptotes.
• The vertices and the covertices are the midpoints of the rectangle
• The covertices are not labeled on the hyperbola because they are not actually part of the graph

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Hyperbola

FroydWess - Online Notes

3/9/2014 10:20 AM

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The following terms are important in drawing the graph of a hyperbola;

Transverse axis is a line segment joining the two vertices of the hyperbola.

Conjugate axis is the perpendicular bisector of the transverse axis.

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General Equations of a Hyperbola

1. Horizontal Transverse Axis : Ax² – Cy² + Dx + Ey + F = 0

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2. Vertical Transverse Axis: Cy² – Ax² + Dx + Ey + F = 0

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3/9/2014 10:20 AM

HYPERBOLA WITH CENTER AT THE ORIGIN C(0,0)

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3/9/2014 10:20 AM

Then letting b² = c² – a² and dividing by a²b², we have

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if foci are on the x-axis

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if foci are on the y-axis

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The generalized equations of hyperbolas with axes parallel to the coordinate axes and center at (h, k) are

if foci are on a axis parallel to

the x-axis

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if foci are on a axis parallel to the y-axis

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3/9/2014 10:37 AM

Theorem Equation of a Hyberbola; Center at (0, 0); Foci at ( +c, 0); Vertices at (+a, 0); Transverse Axis along the x-Axis

An equation of the hyperbola with center at (0, 0), foci at (-c, 0) and (c, 0), and vertices at (- a, 0) and (a, 0) is

The transverse axis is the x-axis.

FroydWess - Online Notes

3/9/2014 10:37 AM

FroydWess - Online Notes

3/9/2014 10:38 AM

Theorem Equation of a Hyberbola; Center at (0, 0); Foci at ( 0, + c); Vertices at (0, + a); Transverse Axis along the y-Axis

An equation of the hyperbola with center at (0, 0), foci at (0, - c) and (0, c), and vertices at (0, - a) and (0, a) is

The transverse axis is the y-axis.

FroydWess - Online Notes

3/9/2014 10:43 AM

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3/9/2014 10:45 AM

Theorem Asymptotes of a Hyperbola

The hyperbola

has the two oblique asymptotes

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3/9/2014 10:47 AM

Theorem Asymptotes of a Hyperbola

The hyperbola

has the two oblique asymptotes

FroydWess - Online Notes

3/9/2014 10:56 AM

Find an equation of a hyperbola with center at the origin, one focus at (0, 5) and one vertex at (0, -3). Determine the oblique asymptotes. Graph the parabola.

Center: (0, 0)

Focus: (0, 5) = (0, c)

Vertex: (0, -3) = (0, -a)

Transverse axis is the y-axis, thus equation is of the form

FroydWess - Online Notes

3/9/2014 1:34 PM

Asymptotes:

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V (0, 3)

V (0, -3)

(4, 0)

(-4, 0)

F(0, 5)

F(0, -5)

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3/9/2014 1:34 PM

Graph:

c² = 9 + 4 = 13

c = √13 = 3.61

Foci: (3.61, 0) and (-3.61, 0)

Center: (0, 0)

The x-term comes first in the subtraction so this is a horizontal hyperbola

Vertices: (2, 0) and (-2, 0)

From the center locate the points that are up three spaces and down three spaces

Draw a dotted rectangle through the four points you have found.

Draw the asymptotes as dotted lines that pass diagonally through the rectangle.

Draw the hyperbola.

From the center locate the points that are two spaces to the right and two spaces to the left

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3/9/2014 1:35 PM

Hyperbola with Transverse Axis Parallel to the x-Axis; Center at (h, k) where b² = c² - a².

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Hyperbola with Transverse Axis Parallel to the y-Axis; Center at (h, k) where b² = c² - a².

FroydWess - Online Notes

3/9/2014 1:35 PM

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Graph:

c² = 9 + 25 = 34

c = √34 = 5.83

Foci: (-7.83, 1) and (3.83, 1)

Center: (-2, 1)

Horizontal hyperbola

Vertices: (-5, 1) and (1, 1)

Asymptotes: y = (x + 2) + 1

53

y = (x + 2) + 1

53

-

FroydWess - Online Notes

3/9/2014 1:35 PM

(y – 1)² (x – 3)² 4 9

c² = 9 + 4 = 13

c = √13 = 3.61

Foci: (3, 4.61) and (3, -2.61)

– = 1

Center: (3, 1)

The hyperbola is vertical

Graph: 9y² – 4x² – 18y + 24x – 63 = 0

9(y² – 2y + ___) – 4(x² – 6x + ___) = 63 + ___ – ___

9

1

9

36

9(y – 1)² – 4(x – 3)² = 36

Asymptotes: y = (x – 3) + 1

23

y = (x – 3) + 1

23

-

FroydWess - Online Notes

3/9/2014 1:35 PM

Center: (-1, -2)

Vertical hyperbola

Find the standard form equation of the hyperbola that is graphed at the right

(y – k)² (x – h)²

b²

a²

– = 1

a = 5 and b = 3

(y + 2)² (x + 1)²

25

9

– = 1

FroydWess - Online Notes

3/9/2014 1:38 PM

Find the center, transverse axis,vertices, foci, and asymptotes of

FroydWess - Online Notes

3/9/2014 1:42 PM

Center: (h, k) = (-2, 4)

Transverse axis parallel to x-axis.

Vertices: (h + a, k) = (-2 + 2, 4) or (-4, 4) and (0, 4)

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Asymptotes:

(h, k) = (-2, 4)

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C(-2,4)

V (-4, 4)

V (0, 4)

F (2.47, 4)

F (-6.47, 4)

(-2, 8)

(-2, 0)

y - 4 = -2(x + 2)

y - 4 = 2(x + 2)

FroydWess - Online Notes

3/9/2014 11:48 AM

Exercise:

1. Find the equation of the hyperbola which satisfies the given conditions

a. Center (0,0), transverse axis along the x-axis, a focus at (8,0), a vertex at (4,0)

b. Center (0,0), transverse axis along the x-axis, a focus at (5,0), transverse axis = 6

c. Center (0,0), transverse axis along y-axis, passing through the points (5,3) and (-3,2).

d. Center (1, -2), transverse axis parallel to the y-axis, transverse axis = 6 conjugate axis = 10

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3/9/2014 10:20 AM

e. Center (-3,2), transverse axis parallel to the y-axis, passing through (1,7), the asymptotes are perpendicular to each other.

f. Center (0,6), conjugate axis along the y-axis, asymptotes are 6x – 5y + 30 = 0 and 6x + 5y – 30 = 0.

2. Reduce each equation to its standard form. Find the coordinates of the center, the vertices and the foci. Draw the asymptotes and the graph of each equation.

a. 9x² –4y² –36x + 16y – 16 = 0

b. 49y² – 4x² + 48x – 98y - 291 = 0

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