Transcript Document

LC.01.4 - The Ellipse
MCR3U - Santowski
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(A) Ellipse Construction

ex 1. Given the circle x2 + y2 = 4 we will apply the
following transformation T(x,y) => (2x,y) which is
interpreted as a horizontal stretch by a factor of 2.

ex 2. Given the circle x2 + y2 = 4, apply the
transformation T(x,y) => (x,3y) which is interpreted as a
vertical stretch by a factor of 3

From these two transformations, we can see that we have
formed a new shape, which is called an ellipse
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(A) Ellipse Construction
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(B) Ellipses as Loci

An ellipse is defined as the set of points such that the sum of the distances
from any point on the ellipse to two stationary points (called the foci) is a
constant

We will explore the ellipse from a locus definition in two ways

ex 3. Using grid paper with 2 sets of concentric circles, we can define the
two circle centers as fixed points and then label all other points (P), that meet
the requirement that the sum of the distances from the point (P) on the ellipse
to the two fixed centers (which we will call foci) will be a constant i.e. PF1 +
PF2 = constant. We will work with the example that PF1 + PF2 = 10 units.

ex 4. Using the GSP program, we will geometrically construct a set of
points that satisfy the condition that PF1 + PF2 = constant by following the
following link
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(B) Ellipses as Loci
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(C) Ellipses as Loci - Algebra

We will now tie in our knowledge of algebra to
come up with an algebraic description of the
ellipse by making use of the relationship that PF1
+ PF2 = constant

ex 5. Find the equation of the ellipse whose foci
are at (+3,0) and the constant (which is called the
sum of the focal radi) is 10. Then sketch the
ellipse by finding the x and y intercepts.
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(C) Ellipses as Loci - Algebra

Since we are dealing with distances, we set up our equation using the
general point P(x,y), F1 at (-3,0) and F2 at (3,0) and the algebra
follows on the next slide |PF1| + |PF2| = 10
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(C) Algebraic Work
PF1  PF2  10



x  32  y 2     x  32  y 2   10



x  3

2
2
y 
 
10 


x  32  y 2
2

x  3
2
y 


2
2
x  32  y 2   x  32  y 2
2
 100 20  x  3  y 2  x 2  6 x  9  y 2
 100  20
x2  6x  9  y2
20

 x  32  y 2
 100  12x
2
2
2

 5  x  3  y 2 
  25  3 x 


25 x 2  6 x  9  y 2  625 150x  9 x 2


25x 2  9 x 2  150x  150x  25 y 2  625 225
16x 2  25 y 2  400
 16x 2   25 y 2

 400 



  400
  x2   y2   x 
 y











  1
  25   16 
 4
 
 
 5
2
2
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(D) Graph of the Ellipse
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(E) Analysis of the Ellipse

The equation of the ellipse is (x/5)2 + (y/4)2 = 1 OR 16x2 + 25y2 = 400
The x-intercepts occur at (+5,0) and the y-intercepts occur at (0,+4)
The domain is {x E R | -5 < x < 5} and the range is {y E R | -4 < y < 4}

NOTE that this is NOT a function, but rather a relation

NOTE the relationship between the equation and the intercepts, domain


and range  so to generalize, if the ellipse has the standard
form
equation (x/a)2 + (y/b)2 = 1, then the x-intercepts occur at

(+a,0), the y-intercepts at (0,+b) and the domain is between –a and +a and
the range is between –b and +b OR we can rewrite the equation in the
form of (bx)2 + (ay)2 = (ab)2
Note that if a > b, then the ellipse is longer along the x-axis than along
the y-axis  so if a < b, then the ellipse would be longer along the y-axis
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(E) Analysis of the Ellipse




The longer of the two axis is called the major axis and lies between the 2 x-intercepts
(if a > b). Its length is 2a
The shorter of the two axis is called the minor axis and lies between the 2 y-intercepts
(if a > b). Its length is 2b
The two end points of the major axis (in this case the x-intercepts) are called vertices
(at (+a,0))
The two foci lie on the major axis
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(F) In-class Examples

Determine the equation of the ellipse and then
sketch it, labelling the key features, if the foci are
at (+4, 0) and the sum of the focal radii is 12 units
(i.e.  the fancy name for the constant distance
sum PF1 + PF2)

The equation you generate should be x2/36 +
y2/20 = 1
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(G) Homework

AW, p470, Q8bc, 9bc
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(H) Internet Links



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http://www.analyzemath.com/EllipseEq/EllipseE
q.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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