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11.3 Ellipses Objective: By the end of the lesson, you should be able to write an equation of an ellipse and sketch its graph.

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Table of Contents Definition of an Ellipse Definition of Foci Definition of Vertices Definition of Major Axis Definition of Minor Axis Definition of Co-vertices Standard Equation of an Ellipse Vertical Major Axis Calculating the Coordinates of the Foci Eccentricity

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Vocabulary you need to know Ellipse – The set of all points (x,y) such that the sum of the distances between (x,y) and two distinct fixed points (foci) are constant. Cabri

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Vocabulary you need to know Foci (plural of focus) - The two distinct points in an ellipse. Focus

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Vocabulary you need to know Vertices – The two points at the intersection of the line through the foci and the ellipse. Cabri

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Vocabulary you need to know Vertices – The two points at the intersection of the line through the foci and the ellipse. Vertex

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Vocabulary you need to know Major Axis – The line segment joining the vertices Major Axis

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Vocabulary you need to know Minor axis – The line segment perpendicular to the major axis at the midpoint. Minor Axis

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Vocabulary you need to know Co-vertices – The endpoints of the minor axis Co-vertex

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In a general ellipse with the center at the origin… a is the distance from the center to the vertex b is the distance from the center to the co-vertex a b Cabri

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In a general ellipse… So the coordinate of the right vertex is (a,0). The coordinate of the left vertex is (-a,0). (-a,0) (a,0) a a 2a Cabri

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In a general ellipse The coordinate of the top co-vertex is (0,b) The coordinate of the bottom vertex is (0,-b) (0,b) (0,-b) b b 2b

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Standard equation of an ellipse with the center at the origin The standard form of the equation of a ellipse with center at (0,0) and major and minor axes of length 2a and 2b, where a>b, is as follows. Horizontal major axis Vertical major axis Cabri

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Standard equation of an ellipse with the center at the origin Horizontal major axis Vertical major axis Notice the switch!!

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What does an ellipse with a vertical major axis look like? Major axis Minor axis Vertex Co-vertex Focus

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How do we calculate the coordinates of the foci? The foci of the ellipse lie on the major axis, c units from the center, where c 2 =a 2 -b 2 Cabri

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How do we calculate the coordinates of the foci? The foci of the ellipse lie on the major axis, c units from the center, where c 2 =a 2 -b 2 c c (c,0) (-c,0)

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What is Eccentricity? Eccentricity tells us how flat (or round) the ellipse is. Cabri

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What is Eccentricity? Eccentricity tells us how flat (or round) the ellipse is. As e approaches 0, the ellipse becomes a circle. As e approaches 1, the ellipse flattens to a line.

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Example 1 Write an equation of an ellipse whose vertices are (-5,0) and (5,0) and whose co-vertices are (0,-3) and (0,3). Find the foci of the ellipse.

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