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COLLEGE ALGEBRA

COLLEGE ALGEBRA. LIAL HORNSBY SCHNEIDER. 6.2. Ellipses. Equations and Graphs of Ellipses Translated Ellipses Ecentricity Applications of Ellipses. Ellipse. An ellipse is the set of all points in a plane the sum of whose distances from

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COLLEGE ALGEBRA

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  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  2. 6.2 Ellipses Equations and Graphs of Ellipses Translated Ellipses Ecentricity Applications of Ellipses

  3. Ellipse An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is constant. Each fixed point is called a focus (plural, foci) of the ellipse.

  4. Equations and Graphs of Ellipses An ellipse has two axes of symmetry, the major axis (the longer one) and the minor axis (the shorter one). The foci are always located on the major axis.

  5. Equations and Graphs of Ellipses The midpoint of the major axis is the center of the ellipse, and the endpoints of the major axis are the vertices of the ellipse. The graph of an ellipse is not the graph of a function.

  6. Equations and Graphs of Ellipses This ellipse has its center at the origin, foci F(c, 0) and F(–c, 0), and vertices V(a, 0) and V(–a, 0). The distance from V to F is a – c and the distance from V to F a + c.

  7. Equations and Graphs of Ellipses The sum of these distances is 2a. Since V is on the ellipse, this sum is the constant referred to in the definition of an ellipse. Thus, for any point P(x, y) on the ellipse,

  8. Equations and Graphs of Ellipses By the distance formula, and

  9. Equations and Graphs of Ellipses Thus, Isolate Be careful when squaring. Square both sides

  10. Equations and Graphs of Ellipses Be careful when squaring. Square x – c: square x + c Isolate

  11. Equations and Graphs of Ellipses Divide each term by 4. Divide by 4. Square both sides. Distributive property

  12. Equations and Graphs of Ellipses Subtract 2ca2x. Rearrange terms. Factor. () Divide by

  13. Equations and Graphs of Ellipses Since B(0, b) is on the ellipse in the figure shown here, we have Combine terms. Divide by 2. Square both sides. Subtract c2.

  14. Equations and Graphs of Ellipses Replacing a2 – c2 with b2 in equation () gives the standard form of the equation of an ellipse centered at the origin with foci on the x-axis. If the vertices and foci were on the y-axis, an almost identical derivation could be used to get the standard form

  15. Standard Forms of Equations for Ellipses The ellipse with center at the origin and equation (a > b) has vertices (a, 0), endpoints of the minor axis (0, b), and foci (c, 0), where c2 = a2 – b2.

  16. Standard Forms of Equations for Ellipses The ellipse with center at the origin and equation (a > b) has vertices (0, a), endpoints of the minor axis (b, 0), and foci (0, c), where c2 = a2 – b2.

  17. GRAPHING ELLIPSES CENTERED AT THE ORIGIN Example 1 Graph each ellipse, and find the coordinates of the foci. Give the domain and range. a. Solution Divide each side of the equation by 36 to get Divide each term by 36. Standard form of an ellipse. Thus, the x-intercepts are  3, and the y-intercepts are  2. The graph of the ellipse is shown in the next slide.

  18. GRAPHING ELLIPSES CENTERED AT THE ORIGIN Example 1 Since 9 > 4, we find the foci of the ellipse by letting a2 = 9 and b2 = 4 in c2 = a2 – b2. (By definition, c > 0.) The major axis is along the x-axis, so the foci have coordinates and The domain of this relation is [–3, 3], and the range is (–2, 2].

  19. GRAPHING ELLIPSES CENTERED AT THE ORIGIN Example 1 Graph each ellipse, and find the coordinates of the foci. Give the domain and range. b. Solution Write the equation in standard form. Divide each term by 64. Standard form of an ellipse. Thus, the x-intercepts are  4, and the y-intercepts are  8. The graph of the ellipse is shown in the next slide.

  20. GRAPHING ELLIPSES CENTERED AT THE ORIGIN Example 1 Here 64 > 16, so a2 = 64 and b2 = 16. Thus The major axis is on the y-axis, so the coordinates of the foci are and . The domain of the relation is the range is [–4, 4]; the range [–8, 8].

  21. Example 1 The graph of an ellipse is not the graph of a function. To graph the ellipse in Example 1(a) with a graphing calculator, solve for y in 4x2 + 9y2 = 36 to get equations of the two functions

  22. WRITING THE EQUATION OF AN ELLIPSE Example 2 Write the equation of the ellipse having center at the origin, foci at (0, 3) and (0, –3), and major axis of length 8 units. Solution Since the major axis is 8 units long, 2a = 8 and a = 4. To find b2, use the relationship a2 – b2 = c2, with a = 4 and c = 3. Substitute for a and c. Solve for b2.

  23. WRITING THE EQUATION OF AN ELLIPSE Example 2 Since the foci are on the y-axis, we use the larger intercept, a, to find the denominator for y2, giving the equation in standard form as

  24. WRITING THE EQUATION OF AN ELLIPSE Example 2 The domain of this relation is and the range is

  25. GRAPHING A HALF-ELLIPSE Example 3 Graph and give the domain and range. Solution Square both sides to get the equation of an ellipse with x-intercepts  5 and y-intercepts  4.

  26. GRAPHING A HALF-ELLIPSE Example 3 Since the only possible values of y are those making giving the half-ellipse shown here. The half-ellipse is the graph of a function. The domain is the interval [– 5, 5] and the range is [0, 4].

  27. Ellipse Centered at (h, k) An ellipse centered at (h, k) with horizontal major axis of length 2a has equation There is a similar result for ellipses having a vertical major axis.

  28. GRAPHING AN ELLIPSE Example 4 Graph and give the domain and range. Solution The graph of this equation is an ellipse centered at (2, –1). Ellipses always have a > b. For this ellipse, a = 4 and b = 3. Since a = 4 is associated with y2, the vertices of the ellipse are on the vertical line through (2, –1).

  29. GRAPHING AN ELLIPSE Example 4 Graph and give the domain and range. Solution Find the vertices by locating two points on the vertical line through (2, –1), one 4 units up from (2, –1), and one 4 units down. The vertices (2, 3) and (2, –5).

  30. GRAPHING AN ELLIPSE Example 4 Graph and give the domain and range. Solution Two other points on the ellipse are on the horizontal line through (2, –1), one 3 units to the right and one 3 units to the left.

  31. GRAPHING AN ELLIPSE Example 4 Graph and give the domain and range. Solution The domain is [–1, 5], and the range is [–5, 3].

  32. Note As suggested by the graphs in this section, an ellipse is symmetric with respect to its major axis, its minor axis, and its center. If a = b in the equation of an ellipse, then the graph is a circle.

  33. Eccentricity The ellipse is the third conic section (or conic) we have studied. (The circle and the parabola were the first two.) The fourth conic section, the hyperbola, will be introduced in the next section. All conics can be characterized by one general definition.

  34. Conic A conic is the set of all points P(x, y) in a plane such that the ratio of the distance from P to a fixed point and the distance from P to a fixed line is constant.

  35. Eccentricity For a parabola, the fixed line is the directrix, and the fixed point is a focus. In the figure shown here, the focus is F(c,0) and the directrix is the line x = –c. The constant ratio is called the eccentricity of the conic, written e. (This is not the same e as the base of natural logarithms.)

  36. Eccentricity If the conic is a parabola, then by definition, the distances d(P, F) and d(P, D) are equal. Thus, every parabola has eccentricity 1.

  37. Eccentricity For an ellipse, eccentricity is a measure of its “roundness.” By the definition of an ellipse, a2 > b2 and and Thus, for the ellipse, Divide by a.

  38. Eccentricity If a is constant, letting c approach 0 would force the ratio to approach 0, which also forces b to approach a (so that would approach 0). Since b determines the endpoints of the minor axis, this means that the lengths of the major and minor axes are almost the same, producing an ellipse very close in shape to a circle when e is very close to 0. In a similar manner, if e approaches 1, then b will approach 0.

  39. Eccentricity The path of Earth around the sun is an ellipse that is very nearly circular. For this ellipse, e .017. On the other hand, the path of Halley’s comet is a very flat ellipse, with e .97. The locations of the foci are shown in each case.

  40. Eccentricity The equation of a circle can be written Divide by r2. In a circle, the foci coincide with the center, so a = b, and and thus e = 0.

  41. APPLYING THE EQUATION OF AN ELLIPSE TO THE ORBIT OF A PLANET Example 5 Find the eccentricity of the ellipse. a. Solution Since 16 > 9, a2 = 16, which gives a = 4. Also, Finally,

  42. APPLYING THE EQUATION OF AN ELLIPSE TO THE ORBIT OF A PLANET Example 5 Find the eccentricity of the ellipse. b. Solution Divide by 50 to obtain Here, a2 = 10, with Now, find c.

  43. APPLYING THE EQUATION OF AN ELLIPSE TO THE ORBIT OF A PLANET Example 6 The orbit of the planet Mars is an ellipse with the sun at one focus. The eccentricity of the ellipse is .0935, and the closest distance that Mars comes to the sun is 128.5 million mi. Find the maximum distance of Mars from the sun. Solution The orbit of Mars has the origin at the center of the ellipse and the sun at one focus. Mars is closest to the sun when Mars is at the right endpoint of the major axis and farthest from the sun when Mars is at the left endpoint. Therefore, the smallest distance is a – c, and the greatest distance is a + c.

  44. APPLYING THE EQUATION OF AN ELLIPSE O THE ORBIT OF A PLANET Example 6 Since a – c = 128.5, c = a – 128.5. Using Multiply by a. Subtract .0935a; add 128.5 Divide by .9065 Then and The maximum distance of Mars from the sun is about 155.1 million mi.

  45. Applications of Ellipses When a ray of light or sound emanating from one focus of an ellipse bounces off the ellipse, it passes through the other focus.

  46. Applications of Ellipses This reflecting property is responsible for whispering galleries. John Quincy Adams was able to listen in on his opponents’ conversations because his desk was positioned at one of the foci beneath the ellipsoidal ceiling and his opponents were located across the room at the other focus.

  47. Applications of Ellipses A lithotripter is a machine used to crush kidney stones using shock waves. The patient is placed in an elliptical tub with the kidney stone at one focus of the ellipse. A beam is projected from the other focus to the tub so that it reflects to hit the kidney stone.

  48. MODELING THE REFLECTIVE PROPERTY OF ELLIPSES Example 7 If a lithotripter is based on the ellipse determine how many units both the kidney stone and the source of the beam must be placed from the center of the ellipse. (c, 0) (– c, 0)

  49. MODELING THE REFLECTIVE PROPERTY OF ELLIPSES Example 7 Solution The kidney stone and the source of the beam must be placed at the foci, (c, 0) and (–c, 0). (c, 0) (– c, 0) Here a2 = 36 and b2 = 27, so

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