Conic Sections Ellipses FCIT Compat

7/28/2019 Conic Sections Ellipses FCIT Compat

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Colleen Beaudoin

January, 2009


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Review: The geometric definition relies on acone and a plane intersecting it

Algebraic definition: a set of points in theplane such that the sum of the distances fromtwo fixed points, calledfoci, remains constant.


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x

y

From eachpoint in theplane, the sumof the distancesto the foci is aconstant.

Example:f1 f2

d2d1

foci

Point A: d1+d2 = c

Point B: d1+d2 = c

BA

d1 d2


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Center

y

f1 f2

foci

x

Major axis

Minor axis


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At your table is paper, corkboard, string,and tacks.

Follow the directions on your handout to

complete the activity.


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Algebraic Definition of an Ellipse: a set of points inthe plane such that the sum of the distances fromtwo fixed points, calledfoci, remains constant.

What remains constant in your sketch? The points where you placed the tacks are known as

the foci. Draw a line through f1 and f2 to the edges ofthe ellipse. This is known as the major axis. Locatethe midpoint between f

1

and f2

. Is this the center ofthe ellipse? Will that always be the case?

What inference can you draw from the data?

Does the data support the definition? Explain.


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Both variables are squared.

Equation:

What makes the ellipse different from thecircle?

What makes the ellipse different from theparabola?


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Procedure to graph:

1. Put in standard form (above): x squared term +y squared term = 1

2. Plot the center (h,k)

3. Plot the endpoints of the horizontal axis bymoving a units left and right from thecenter.

2 2

2 2

( - ) ( - )1

x h y k

a b


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To graph:

4. Plot the endpoints of the vertical axis bymoving b units up and down from thecenter.

Note: Steps 3 and 4 locate the endpoints of the major

and minor axes.

5. Connect endpoint of axes with smooth curve.

2 2

2 2

( - ) ( - )1

x h y k

a b


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To graph:

6. Use the following formula to help locate thefoci: c2 = a2 - b2 if a>b or c2 = b2 a2 if b>a

**Challenge question: Why are we using thisformula to locate the foci? Draw a diagramand justify your answer.**

2 2

2 2

( - ) ( - )1

x h y k

a b


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To graph:

6. (continued) Move c units left and right formthe center if the major axis is horizontal

OR Move c units up and down form the centerif the major axis is vertical

Label the points f1 and f2 for the two foci.

Note: It is not necessary to plot the foci to graph the ellipse, but it is commonpractice to locate them.

2 2

2 2

( - ) ( - )1

x h y k

a b


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To graph:

7. Identify the length of the major and minoraxes.

2 2

2 2

( - ) ( - )1

x h y k

a b


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To graph:

1. Put in standard form (set = 1)

Done2. Plot the center (h,k)

(-2,3)

3. Plot the endpoints of the horizontal axis by

moving a units left and right from thecenter.

Endpoints at (-7,3) and (3,3)

2 2( 2) ( - 3)1

25 16

x y


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4. Plot the endpoints of the vertical axis bymoving b units up and down from thecenter.

Endpoints at (-2,7) and (-2,-1)

5. Connect endpoint of axes with smooth curve

2 2( 2) ( - 3)1

25 16

x y


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Major axis

Center

Minor axis

2 2( 2) ( - 3)1

25 16

x y


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6. Which way is the major axis in this problem (horizontalor vertical)?

Horizontal because 25>16 and 25 is under the xUse the following formula to help locate the foci: c2 = a2 - b2

if a>b or c2 = b2 a2 if b>ac2 = a2 - b2

c2 = 25 16c2 = 9

c

= 3Move 3 units left and right from the center to locate thefoci.

Where are the foci?(-5,3) and (1,3)

2 2( 2) ( - 3)1

25 16

x y


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Foci

f1

f2

Length of Major Axis is 10.Length of Minor Axis is 8.

2 2( 2) ( - 3)1

25 16

x y


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To graph:

1. Put in standard form.

2. Plot the center

(0,0)

3. Plot the endpoints of the horizontal axis.


2 2

19 16

x y


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4. Plot the endpoints of the vertical axis.Endpoints at (0,4) and (0,-4)


6. Which way is the major axis in this problem?Vertical because 16>9 and 16 is under the yLocate the foci:c2 = b 2 - a2

c2 = 16 - 9

c2 = 7c= 7Where are the foci?(0, 7) and (0,-7)

2 2

1

9 16

x y

2 2


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2 2

19 16

x y

Length of Major Axis is 8.Length of Minor Axis is 6.


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1. Put in standard form.(Hint: Complete the square.)

4x2 + 16x + 9y2 54y = -614(x2 + 4x ) + 9(y2 6y ) = -61

+4 +9 +16 + 814(x + 2)2 + 9(y 3)2 = 36

2. Plot the center(-2,3)

3. Plot the endpoints of the horizontal axis.


2 2( 2) ( 3)1

9 4

x y


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4. Plot the endpoints of the vertical axis.Endpoints at (-2,5) and (-2,1)


6. Which way is the major axis in this problem?HorizontalLocate the foci:c2 = a2 - b2

c2 = 9 - 4

c2 = 5c= 5Where are the foci?(-2 5, 3)

2 2( 2) ( 3)1

9 4

x y


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Length of Major Axis is 6.

Length of Minor Axis is 4.

2 2( 2) ( 3)1

9 4

x y


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Given the following information, write theequation of the ellipse. Sketch and find thefoci.

Center is (4,-3), the major axis is vertical andhas a length of 12, and the minor axis has alength of 8.


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1) How can you tell if the graph of an equationwill be a line, parabola, circle, or an ellipse?

2) Whats the standard form of an ellipse?3) What are the steps for graphing an ellipse?4) Whats the standard form of a parabola?5) Whats the standard form of a circle?

6) How are the various equations similar anddifferent?

Conic Sections Ellipses FCIT Compat

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