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11.3 Ellipses

11.3 Ellipses. ©2001 by R. Villar All Rights Reserved. Ellipses. Sum of the distances: 12 units. co-vertex. foci. vertex. vertex. Ellipse: set of all points in a plane such that the sum of the distances from two given points in a plane, called the foci , is constant. co-vertex.

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11.3 Ellipses

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  1. 11.3 Ellipses ©2001 by R. Villar All Rights Reserved

  2. Ellipses Sum of the distances: 12 units co-vertex foci vertex vertex Ellipse: set of all points in a plane such that the sum of the distances from two given points in a plane, called the foci, is constant. co-vertex The major axis is the line segment joining the vertices (through the foci) The minor axis is the line segment joining the co-vertices (perpendicular to the major axis)

  3. This is the equation if the major axis is horizontal. Standard Equation of an Ellipse (Center at Origin) (0, b) (–c, 0) (c, 0) (–a, 0) (a, 0) x2 + y2 = 1 a2 b2 (0, –b) The foci of the ellipse lie on the major axis, c units from the center, where c2 = a2 – b2

  4. This is the equation if the major axis is vertical. Standard Equation of an Ellipse (Center at Origin) (0, a) (0, c) (–b, 0) (b, 0) (0, –c) x2 + y2 = 1 b2 a2 (0, –a) The foci of the ellipse lie on the major axis, c units from the center, where c2 = a2 – b2

  5. Since the major axis is vertical, the equation is the following: Example: Write an equation of the ellipse whose vertices are (0, –3) and (0, 3) and whose co-vertices are (–2, 0) and (2, 0). Find the foci of the ellipse. (0, 3) (0, c) (–2, 0) (b, 0) Since a = 3b = 2 The equation isx2 + y2 = 1 4 9 (0, –c) (0, –3) x2 + y2 = 1 b2 a2 Use c2 = a2 – b2 to find c. c2 = 32 – 22 c2 = 9 – 4 = 5 c = The foci are

  6. Example: Write the equation in standard form of 9x2 + 16y2 = 144. Find the foci and vertices of the ellipse. Get the equation in standard form (make it equal to 1): 9x2 + 16y2 = 144 144 144 144 Simplify... x2 + y2 = 1 16 9 That means a = 4b = 3 Use c2 = a2 – b2 to find c. c2 = 42 – 32 c2 = 16 – 9 = 7 c = (0, 3) (–4,0) (4, 0) Vertices: Foci: (–c,0) (c, 0) (0,-3)

  7. Eccentricity is given by e = c a Eccentricity of an ellipse is a measure of its ovalness... The closer the eccentricity is to 1, the more elongated it is. The closer the eccentricity is to 0, the more circular it is. (A circle has an eccentricity of 0) Find the eccentricity of the ellipse below: a = 5b = 3 c = 4 (0, 3) e = c a e = 45 e = 0.8 (–5,0) (–4,0) (4, 0) (5, 0) (0,-3)

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