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1 LC.01.4 - The Ellipse MCR3U - Santowski

1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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Page 1: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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LC.01.4 - The Ellipse

MCR3U - Santowski

Page 2: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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

Page 3: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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(A) Ellipse Construction

Page 4: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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

Page 5: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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(B) Ellipses as Loci

Page 6: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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.

Page 7: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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

Page 8: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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(C) Algebraic Work

1451625400

25400

16

4002516

22562525150150925

91506259625

32535

12100320

9632010096

33201003

3103

1033

10

222222

22

222

222

22

22

22

222222

222222

222

222

2222

21

yxyxyx

yx

yxxxx

xxyxx

xyx

xyx

yxxyxyxx

yxyxyx

yxyx

yxyx

PFPF

Page 9: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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(D) Graph of the Ellipse

Page 10: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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

Page 11: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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

Page 12: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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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

Page 13: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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(G) Homework AW, p470, Q8bc, 9bc

Page 14: 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y 2 = 4 we will apply the following transformation

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(H) Internet Links 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