1 LC The Ellipse (Algebraic Perspective) MCR3U - Santowski

2 (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)  (Alternatively, if the foci are on the y-axis (and the major axis is on the y-axis), then the equation becomes x 2 /b 2 + y 2 /a 2 = 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 c 2 = a 2 – b 2

3 (B) Translating Ellipses  So far, we have considered ellipses from a geometric perspective |PF 1 + PF 2 | = 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 y 2 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 5 2 – 4 2 = +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 c 2 = a 2 – b 2 = 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  q.html - an interactive applet fom AnalyzeMath q.html q.html  - Examples and explanations from OJK's Precalculus Study Page  Ellipses.asp - Ellipses from Paul Dawkins at Lamar University Ellipses.asp Ellipses.asp  - 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