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LC.02.1 - The Ellipse (Algebraic Perspective)

LC.02.1 - The Ellipse (Algebraic Perspective). MCR3U - Santowski. (A) Review. The standard equation for an ellipse is x 2 /a 2 + y 2 /b 2 = 1 (where a>b and the ellipse has its foci on the x-axis and where the major axis is on the x-axis)

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LC.02.1 - The Ellipse (Algebraic Perspective)

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  1. LC.02.1 - The Ellipse (Algebraic Perspective) MCR3U - Santowski

  2. (A) Review • The standard equation for an ellipse is x2/a2 + y2/b2 = 1 (where a>b and the ellipse has its foci on the x-axis and where the major axis is on the x-axis) • (Alternatively, if the foci are on the y-axis (and the major axis is on the y-axis), then the equation becomes x2/b2 + y2/a2 = 1, where b>a) • The intercepts of our ellipse are at +a and +b • The vertices of the ellipse are at +a and the length of the major axis is 2a • The length of the minor axis is 2b • The domain and range can be determined from the values of a and b and knowing where the major axis lies • The two foci are located at (+c,0) or at (0,+c) • NEW POINT  the foci are related to the values of a and b by the relationship that c2 = a2 – b2

  3. (B) Translating Ellipses • So far, we have considered ellipses from a geometric perspective |PF1 + PF2| = 2a and we have centered the ellipses at (0,0) • Now, if the ellipse were translated left, right, up, or down, then we make the following adjustment on the equation:

  4. (C) Translating Ellipses – An Example • Given the ellipse determine the center, the vertices, the endpoints of the minor axis, the foci, the intercepts. Then graph the ellipse. • The center is clearly at (3,-4)  so our ellipse was translated from being centered at (0,0) by moving right 3 and down 4  so all major points and features on the ellipse must also have been translated R3 and D4 • Since the value under the y2 term is greater (25>16), the major axis is on the y-axis, then the value of a = 5 and b = 4 • So the original vertices were (0,+5) and the endpoints of the minor axis were (+4,0)  these have now moved to (3,1), (3,-9) as the new vertices and (-1,-4) and (7,-4) as the endpoints of the minor axis • The original foci were at 52 – 42 = +3  so at (0,+3) which have now moved to (3,-1) and (3,-7)

  5. (C) Translating Ellipses – The Intercepts • For the x-intercepts, set y = 0 and for the y-intercepts, set x = 0

  6. (C) Translating Ellipses – The Graph

  7. (D) In-Class Examples • Ex 1. Graph and find the equation of the ellipse whose major axis has a length of 16 and whose minor axis has a length of 10 units. Its center is at (2,-3) and the major axis is parallel to the y axis • So 2a = 16, so a = 8 • And 2b = 10, thus b = 5 • And c2 = a2 – b2 = 64 – 25 = 39  c = 6.2 • Therefore our non-translated points are (0,+8), (+5,0) and (0,+6.2)  now translating them by R2 and D3 gives us new points at (2,5),(2-11),(-3,-3),(7,-3),(2,3.2),(2,-9.2) • Our equation becomes (x-2)2/25 + (y+3)2/64 = 1

  8. (D) In-Class Examples

  9. (E) Internet Links • http://www.analyzemath.com/EllipseEq/EllipseEq.html - an interactive applet fom AnalyzeMath • http://home.alltel.net/okrebs/page62.html - Examples and explanations from OJK's Precalculus Study Page • http://tutorial.math.lamar.edu/AllBrowsers/1314/Ellipses.asp - Ellipses from Paul Dawkins at Lamar University • http://www.webmath.com/ellipse1.html - Graphs of ellipses from WebMath.com

  10. (F) Homework • AW text, page 528-9, Q2,4d,5d,8,9 • Nelson text, p591, Q2eol,3eol,5,8,10,11,15,16

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