table of contents ellipse - finding the equation recall that the two equations for the ellipse are...
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Table of Contents
Ellipse - Finding the Equation
• Recall that the two equations for the ellipse are given by ...
1b
)ky(
a
)hx(2
2
2
2
Horizontal Ellipse Vertical Ellipse
1a
)ky(
b
)hx(2
2
2
2
Table of ContentsSlide 2
Ellipse - Finding the Equation
• Summary of meanings of a, b, h, and k:
The end points of the minor axis are b units from the center.
The vertices are a units from the center on the major axis.
The center is at C(h, k).
The foci are c units from the center on the major axis.
Table of ContentsSlide 3
Ellipse - Finding the Equation
• Example 1: Find the equation of the ellipse with the given vertices and foci ...
)6,3(F),2,3(F
)8,3(V),4,3(V
21
21
• The first step is to plot these points on a coordinate plane.
Table of ContentsSlide 4
Ellipse - Finding the Equation
• The center will be the midpoint of the line segment joining the vertices (also the foci) ...
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -6 -4 -2 2
)6,3(F2
)2,3(F1
)8,3(V2
)4,3(V1
)2,3(C • The distance from the center to a vertex is a = 6.
• The distance from the center to a focus is c = 4.
Table of ContentsSlide 5
Ellipse - Finding the Equation
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -6 -4 -2 2
)6,3(F2
)2,3(F1
)8,3(V2
)4,3(V1
)2,3(C
• Using the formula ...222 bac
52b
20b
1636b
b64
2
2
222
the value of b can be found.
Table of ContentsSlide 6
Ellipse - Finding the Equation
• Summary:
52b,6a
)2,3(C:Center
• Since the vertices are vertical, the ellipse is vertical ...
1a
)ky(
b
)hx(2
2
2
2
and the equation is ...
136
)2y(
20
)3x( 22
Table of ContentsSlide 7
Ellipse - Finding the Equation
• Example 2: Find the standard form of the ellipse given by ...
047y32x54y16x9 22 • Solution:
47)y32y16()x54x9( 22
047y32x54y16x9 22
47)y2y(16)x6x(9 22
168147)1y2y(16)9x6x(9 22
Table of ContentsSlide 8
Ellipse - Finding the Equation
168147)1y2y(16)9x6x(9 22
144)1y(16)3x(9 22
144
144
144
)1y(16
144
)3x(9 22
19
)1y(
16
)3x( 22
• The standard form of the ellipse is ...
Table of Contents
Ellipse - Finding the Equation