LC.02.1 - The Ellipse
(Algebraic Perspective)
MCR3U - Santowski
1
(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
2
(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:
x h
a
2
2
y k
b
2
2
1
3
(C) Translating Ellipses – An Example
x 3
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)
4
(C) Translating Ellipses – The Intercepts
For the x-intercepts, set y = 0 and for the y-intercepts, set x = 0
(0 3) 2 ( y 4) 2
1
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
5
(C) Translating Ellipses – The Graph
6
(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
7
(D) In-Class Examples
8
(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
9
(F) Homework
AW
text, page 528-9, Q2,4d,5d,8,9
Nelson
text, p591,
Q2eol,3eol,5,8,10,11,15,16
10
0
