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10.3: Ellipses

Content Objective: I will be able to write the standard equation for an ellipse, graph an ellipse, and identify its center, vertices, co­vertices, and foci.

Vocabulary

Co­vertices (ellipse): the endpoints of the minor axis.

Focus (ellipse): one of the two fixed points F1 and F2 that are used to define an ellipse. For every point P on the ellipse, PF1+PF2 is constant

Major Axis: the longer axis of an ellipse. The foci of the ellipse are located on the major axis, and its endpoints are the vertices

Vertices (ellipse): The endpoints of the major axis of the ellipse.

Minor Axis (ellipse): the shorter axis of an ellipse

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Notes:

If you pulled the center of a circle apart into two points called the foci, it would stretch the circle into an ellipse.

An ellipse is the set of points such that the sum of the distances from any point on the ellipse to the foci is a constant.

Horizontal Ellipse in Standard Form:

Vertical Ellipse in Standard Form:

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To find the foci (c­value)

c=√(a2­b2)

where a is your major axis and b is your minor axis

If your ellipse is horizontal

(h±c,k)If your ellipse is vertical

(h,k±c)

The points on your graph for your foci are:

Remember your foci are located on your major axis only

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To find the vertices

Your vertices will depend on your major and minor axis:

a­ major axis

b­ minor axis.

Recall that the center is the point (h,k)

Your vertices on the x ­ axis you will be:

If your ellipse is horizontal

(h±a, k)

If your ellipse is vertical

(h,k±a)

Your vertices on the y ­ axis you will be:

If your ellipse is horizontal

(h, k±b)

If your ellipse is vertical

(h±b,k)

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Ex 1: Graphing an Ellipse

Graph and identify the center, , foci, vertices, and co­vertices

A)

2) Identify type of ellipse: if major axis is the x axis, it is horizontal. If major axis is y­axis, it is vertical

3) Identify Center

1) Identify major/minor axis

4) Identify the vertices and foci:

For x­axis (vertex): (h ± a, k)

For y­axis (co­vertex): (h, k ± b)

Foci:

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A)

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B)

Try it (worksheet)

C)

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The standard form of an equation of an ellipse with center at the origin is:

Recall that a is your major axis, and b is your minor axis. In the case show above, the x­axis is the major axis.

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Ex 2: Writing an equation in standard form

Write an equation in standard form for each ellipse with center (0, 0)

A) Vertex at (6, 0); co­vertex at (0, 4)

B) Vertex at (0, 5); co­vertex at (2, 0)

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Try It (Worksheet)

Write an equation in standard form for each ellipse with center (0, 0)

A) Vertex at (0, ­6); co­vertex at (3, 0)

B) Vertex at (9, 0); co­vertex at (0, 5)

If the equation is not is standard form, try dividing by the constant.

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Ex 3: Find the Foci of an Ellipse

A) Find the foci of the ellipse with the equation . Graph the ellipse.

1) write in standard form

c=√(a2­b2)recall the eq for finding the foci

3) solve for c

2) identify major axis

note: in the equation above 25 is a2 and 9 is b2. so we do not need to square these values

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Try It: Worksheet

Find the foci of the ellipse with the equation . Graph theellipse.

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EX 4: Using the foci of an ellipse

Write an equation of the ellipse with foci at (±7, 0) and co­vertices at (0, ±6)

1) identify major axis

2) use the foci and co­vertices to find a and b

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Try It: Worksheet

Write an equation of the ellipse with foci at (0, ±4) and co­vertices at (±8, 0)

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Fin

HW: Worksheet due Thursday 5/1