LC.02.1 - The Ellipse
(Algebraic Perspective)
MCR3U - Santowski
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(A) Review
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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
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(B) Translating Ellipses
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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:
 x  h
a
2
2

y  k
b
2
2
1
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(C) Translating Ellipses – An Example
 x  3
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2

 y  4
2
1
Given the ellipse 16
determine the center,
25
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)
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(C) Translating Ellipses – The Intercepts
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For the x-intercepts, set y = 0 and for the y-intercepts, set x = 0
(0  3) 2 ( y  4) 2
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16
25
25(9)  16( y  4) 2  400
400  225
( y  4) 
 10.9375
16
y  4   3.307.....
2
  7.3
y 
  0.7
( x  3) 2 (0  4) 2

1
16
25
25( x  3) 2  16(16)  400
400  256
( x  3) 
 5.76
25
x  3  2.4
2
 0.6
x 
 5.4
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(C) Translating Ellipses – The Graph
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(D) In-Class Examples
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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
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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
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(D) In-Class Examples
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(E) Internet Links
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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
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(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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