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Ellipses and Circles

Ellipses and Circles. Section 10.3. 1 st Definition. An ellipse is a conic section formed by a plane intersecting one cone not perpendicular to the axis of the double-napped cone.

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Ellipses and Circles

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  1. Ellipses and Circles Section 10.3

  2. 1st Definition

  3. An ellipse is a conic section formed by a plane intersecting one cone not perpendicular to the axis of the double-napped cone. • A circle is a conic section formed by a plane intersecting one cone perpendicular to the axis of the double-napped cone.

  4. 2nd Definition

  5. An ellipse is the set of all points (x, y) in a plane, • the sum of whose distances from two distinct • fixed points (foci) is constant. • d1 + d2 = constant Turn on N-Spire Calculator. Open the file Ellipse Construction. d1 d2

  6. The line through the foci intersects the ellipse • at two points, called vertices. The chord joining • the vertices is the major axis, and its midpoint is • the center of the ellipse. The chord perpendicular • to the major axis at the center is the minor axis • of the ellipse. minor axis major axis center vertex vertex

  7. General Equation of an Ellipse • Ax2 + Cy2 + Dx + Ey + F = 0 • If A = C, then the ellipse is a circle.

  8. Standard Equation of an Ellipse

  9. The standard form of the equation of an ellipse, with center (h, k) and major and minor axes of lengths 2a and 2b respectively, where 0 < b < a, where the major axis is horizontal. where the major axis is vertical.

  10. The foci lie on the major axis, c units from the center, with c2 = a2 – b2. • What is true about c in a circle? Why? • It is equal to 0. • Because a and b are equal lengths. • What is true about the center and foci of a circle? • They are all the same point. • To measure the ovalness of an ellipse, you can use the concept of eccentricity.

  11. Eccentricity ofan Ellipse

  12. The eccentricity of an ellipse is given by the ratio • Note that 0 < e < a for every ellipse. Why? • c < a • The closer that the eccentricity is to 1 the more elongated the ellipse. • What is the eccentricity of a circle? • 0

  13. Examples • For the following ellipse, find the center, a, b, c, vertices, the endpoints of the minor axis, foci, eccentricity, and graph. • What must you do to the above equation to do this example? • Complete the square twice. 16x2 + y2 − 64x + 2y + 49 = 0

  14. center: (2, -1) a = 4 b = 1 What type of ellipse is this ellipse? vertical ellipse?

  15. vertices: • endpoints of the minor axis: • eccentricity: • foci: (2, 3), (2, −5) (3, −1), (1, −1)

  16. V1 F1 C F2 V2

  17. A circle is a special ellipse. The center and the two foci are the same point. • A circle is a set of points in a plane a given distant from a given point. • The standard form of the equation of a circle with center (h, k) and radius, ris • (x – h)2 + (y – k)2= r2

  18. Example • Find the standard form of the equation of the circle, center, radius and graph.

  19. (x + 4)2 + (y – 9)2 = 25 • center: (-4, 9) • radius = 5 END OF THE 1ST DAY

  20. Each focus has its own line that relates to the ellipse, this line is called the directrix. If we have an ellipse with a major axis distance of aand an eccentricity of e, the directrix of the ellipse is defined as the lines perpendicular to the line containing the major axis at a distance from the

  21. Because the eccentricity of an ellipse is positive and less than 1, we know that • and therefore we know that the directrix does not intersect the ellipse. Why?

  22. Directrix Directrix a

  23. Focus Directrix Property of Ellipses • This property explains how the directrix relates to an ellipse. This is the 3RD DEFINITION OF AN ELLIPSE.

  24. An ellipse is the set of all points P such that the distance from a point on the ellipse to the focus F is e times the distance from the same point to the associated directrix. F1P= e • AP d2 P A d1 F1 Directrix Directrix

  25. Example • Given: a vertical ellipse with • Find the length from P(2, 4), a point on the ellipse to the focus associated with the given directrix.

  26. F1P= e • AP

  27. Examples • Write the equation of each ellipse described. Find the equation of each directrix. Graph.

  28. Center (0, 0), a= 6, b = 4 horizontal major axis. • To find the equation of the directrix. • Find c. • b. Find e.

  29. Center (6, 1), foci (6, 5) and (6, −3) • length of major axis is 10 • vertical major axis • 2c = 5 + 3 = 8 so c = 4 • 16 = 25 – b2 • b2 = 9

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