1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y...
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Transcript of 1 LC.01.4 - The Ellipse MCR3U - Santowski. 2 (A) Ellipse Construction ex 1. Given the circle x 2 + y...
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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
1451625400
25400
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4002516
22562525150150925
91506259625
32535
12100320
9632010096
33201003
3103
1033
10
222222
22
222
222
22
22
22
222222
222222
222
222
2222
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yxyxyx
yx
yxxxx
xxyxx
xyx
xyx
yxxyxyxx
yxyxyx
yxyx
yxyx
PFPF
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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 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