Obj. 29 Ellipses (Presentation)

7/31/2019 Obj. 29 Ellipses (Presentation)

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Obj. 29 Ellipses

Unit 7 Conic Sections


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Concepts and Objectives

Ellipses (Obj. 29)

Identify the equation of an ellipse Find the center,x-radius, andy-radius of an ellipse

Find the major and minor axes

Find the foci and focal length of an ellipse Write the equation of an ellipse

Solve problems involving ellipses


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Ellipses

An ellipse, geometrically speaking, is a set of points in a

plane such that for each point, thesum of its distances,d1 + d2, from two fixed points F1 and F2, is constant.

What does this mean?

Put a piece of paper on top of the cardboard.

Place the two pins at least 3" apart.

Tie your piece of string in a loop that will fit around

the pins without going off the edge.

Put the loop around the pins, pull it taut, and trace

around the loop.


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Parts of an Ellipse

Parts of an ellipse:

Center Vertices (ea. vertex)

Major axis

Minor axis Foci (ea. focus)

What we have done with the string is kept the distancebetween the foci and the points on the ellipse constant

(i.e. the definition).


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Parts of an Ellipse

Other important parts:

Thesemi-majorandsemi-minoraxes are half thelength of the major and minor axes.

The distance from the center to the ellipse in thex-

direction is called thex-radius. Likewise, the distance

in they-direction is called they-radius.

The distance between the foci is called thefocal

length. The distance between the center and a focus

is called thefocal radius.


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Ellipses

With ellipses, we use the following notation:

a is the length of the semi-major axis b is the length of the semi-minor axis

c is the length of the focal radius.

Therefore,

The length of the major axis is 2a

The length of the minor axis is 2b The sum of the distances from a point(x,y) on the

ellipse to the two foci is 2a


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Focal Radius

Not only does the dotted

line trace the outline of theellipse, but it is also the

length of the major axis. As

you can see from the

picture, half of that length(a) is the hypotenuse of the

triangle formed by the semi-

minor axis and the focal

radius. This gives us the

formula:=

2 2 2c a b

ba

c

(careful!)


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Standard Form

The standard form of an ellipse centered at(h, k) is

where rxis thex-radius and ryis they-radius. To graph an ellipse from the standard form, plot the

center, mark thex- andy-radii, and sketch in the curve.

+ =

22

1

x y

x h y k

r r


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Standard Form

Example: Sketch the graph of

The center is at(4, 1)Thex-radius is 3

They-radius is 5

Sketch in the curves

+ + =

2 24 11

3 5

x y


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Standard Form of an Ellipse

Example: Sketch the graph of

To sketch the graph, we have to rewrite the equation:

+ + + =2 24 9 16 90 205 0x y x y

( ) ( ) + + = 2 24 16 9 90 205x x y y

( ) ( ) ( ) ( ) + + = + ++ +2 2222 24 4 10 2052 4 29 5 9 5x x y y

( ) ( ) ++ =

2 2

4 2 9 5 36

36 36 36

x y

+ + =

2 2

2 51

3 2

x y( ) ( ) ++ =

2 2

2 51

9 4

x y

Remember, you

have to factor out

the coefficient ofx2!


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Standard Form of an Ellipse

Example (cont.):

The center is at(2, 5)Thex-radius is 3

They-radius is 2

+ + =

2 2

2 51

3 2

x y


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Focal Length

Example: Write the equation of the ellipse having center

at the origin, foci at(5, 0) and (5, 0), and major axis oflength 18 units.

Since the major axis is 18 units long, 2a = 18, so a = 9.

The distance between the center and a focus, c, is 5.

Therefore, we can find b using the formula:

= 2 2 2c a b

=

2 2 2

5 9 b= =

2 2 29 5 56b

= 56b


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Focal Length

Example (cont.):

The foci lie along the major axis, so we know thatrx= a.

Putting it all together, we have:

or

+ =

22

19 56

x y

+ =

2 2

1

81 56

x y


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Eccentricity The eccentricity of an ellipse is a measure of its

roundness, and it is the ratio of the focal length to themajor axis.

This ratio is written as

=c

e a


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Eccentricity Example: The orbit of Jupiter is an ellipse with the sun

at one focus (mostly). The eccentricity of the ellipse is0.0489, and the maximum distance of Jupiter from the

sun is 507.4 million miles. Find the closest distance that

Jupiter comes to the sun.

Jupiter

Sun

a + c

a c= 0.0489

c

a

= 0.0489c a

+ = 507.4

a c+ =0.0489 507.4a a

= 507.4

483.741.0489

a


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Eccentricity Example (cont.):

Jupiter

Sun

a + c

a c

( )= 0.0489 483.74c

= 23.65c

= 483.74 23.65a c

= 460.1 million miles


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Homework College Algebra

Page 968: 15-39 (3s), 45, 47a, 48 HW: 18, 24, 27, 39, 45, 47a

Obj. 29 Ellipses (Presentation)

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