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**Analytic Geometry Section 3.3**

The Ellipse Analytic Geometry Section 3.3

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**Definition of “ellipse”**

An ellipse is the set of all points in a plane such that the distance from two fixed points (foci) on the plane is a constant.

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**Equation of the Ellipse**

The equation of an ellipse with its center at the origin has one of two forms: The position of the a2 (under the x or y) tells you whether the horizontal or the vertical axis is the major axis of the ellipse.

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Ellipse This ellipse has a horizontal major axis that is 16 units long.

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Ellipse The minor axis of this ellipse is 10 units in length.

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Foci The two foci for this ellipse are the two points lying on the horizontal axis that appear to be a little over 6 units from the origin. The origin is the center of the ellipse. The distance from the center to a focus is “c”.

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The segments drawn from the two foci to the point (0,5) on the ellipse are each 8 units in length. Their total length is 16 units. This total length is also the length of the major axis.

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**Two more segments are added, drawn from the foci to the point (2,4**

Two more segments are added, drawn from the foci to the point (2,4.84) on the ellipse. Their lengths are and The sum of these lengths is again 16 units.

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**The two latest segments, drawn to the point (7,-2**

The two latest segments, drawn to the point (7,-2.42) on the ellipse, are units and units in length, a sum of 16 units.

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**The Ellipse The ends of the major axis are at (a,0) and (-a,0).**

The ends of the minor axis are at (0,b) and (0,-b). The foci are at (c,0) and (-c,0).

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**The Ellipse The sum of the distances from point P to the foci is 2a.**

Also,

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**The endpoints of the two latus recti are found using the equivalence :**

A chord through a focus and perpendicular to the major axis is called a latus rectum. The endpoints of the two latus recti are found using the equivalence :

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**Latus Rectum When the equation of the ellipse is**

So the endpoints of the latus recti are:

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The Ellipse The major axis is the vertical axis with endpoints (0,13) and (0,-13). The endpoints of the major axis are called the vertices. The minor axis has endpoints of (5,0) and (-5,0).

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**The foci are found using**

so the values of c are 12 and -12. The coordinates of the foci are (0,12) and (0,-12).

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**The endpoints of the latus recti are:**

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**Problem: Determining an equation**

Find the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). First, place these points on axes. The F and F’ are the foci.

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**The values of a and b need to be determined.**

Finding the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). Since the vertex is on the horizontal axis, the ellipse will be of the form: The values of a and b need to be determined.

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**applies. Solve for b2 to get In this case,**

Finding the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). If the foci are at 8 and -8, then c = 8. Since a vertex is at (12,0), that means that a = 12. Relating these values to the standard form for an ellipse whose center is at the origin and whose major axis is horizontal, , and the equivalence applies. Solve for b2 to get In this case,

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**So the equation of the ellipse is: or**

Finding the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). Since The value of a is 12, and a2 is 144. The value of b is and b2 is 80. So the equation of the ellipse is: or

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**Ellipse with center at (h,k)**

The ellipses with their centers at the origin are just special cases of the more general ellipse with its center at the point (h,k). This more general ellipse has a standard formula of:

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**Problem: Write the equation in standard form**

The general form of the equation is: After writing this in standard form, also find the coordinates of the center, the foci, the ends of the major and minor axes, and the ends of each latus rectum.

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**Write in standard form:**

First, group the terms with x’s and the terms with y’s, and move the constant to the other side of the equation.

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**Write in standard form:**

Now factor out the coefficient of each squared term. Then complete the square for each variable.

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**To finish the problem: Simplify on the right.**

Then divide each side by 32.

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**The Ellipse This ellipse has a center at .**

The major axis is in length, and the minor axis is 4 in length, so their endpoints are ( ,0) and (- ,0), (0,2) and (0,-2). The foci are at (2,0) and (-2,0).

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To finish Since the foci are at (2,0) and (-2,0), the endpoints of the latus recti are at

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Site and Assignment There’s a neat website that you might want to look at for more on the ellipse. It’s at: Your assignment, due Monday, is: 3.3: 2, 3, 15, 16, 17, 22, 25, 44

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