warm up write the standard form of the equation: then find the radius and the coordinates of the...
DESCRIPTION
Ellipse - Definition ellipse foci An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant. d 1 + d 2 = a constant value. center The center of the ellipse is the midpoint of the line segment that joins the fociTRANSCRIPT
![Page 1: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/1.jpg)
Warm up
•Write the standard form of the equation:
• Then find the radius and the coordinates of the center.
• Graph the equation
0132822 yxyx
![Page 2: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/2.jpg)
Lesson 10-3 EllipsesObjective: To use and determine the standard and general forms of the equation of an ellipseTo graph ellipses
![Page 3: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/3.jpg)
Ellipse - DefinitionAn ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant.
d1 + d2 = a constant value. The center of the ellipse is the midpoint of the
line segment that joins the foci
![Page 4: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/4.jpg)
An ellipse has 2 axes of symmetry. The longer one contains the foci and is the major axis. The shorter one is called the minor axis.
F1: (-c,0) F2: (c,0)
Minor Axis
Major Axis
Vertex Vertex
Vertex
Vertex
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Ellipse - Equation
2 2
2 2 1x h y ka b
The equation of an ellipse centered at (0, 0) is ….
2 2
2 2 1x ya b
where c2 = a2 – b2 andc is the distance from the center to the foci.
Shifting the graph over h units and up k units, the center is at (h, k) and the equation is
where c2 = a2 – b2 andc is the distance from the center to the foci.
![Page 6: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/6.jpg)
Equation of an Ellipse: Center at (0,0); Foci at (0, c) and (0, -c); Major Axis is Vertical
12
2
2
2
ay
bx where a2 ≥ b2 and
b2 = a2 - c2
The major axis is the y - axis. The vertices are at (0, -a) and (0, a)
![Page 7: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/7.jpg)
aab
b cc
Ellipse - Graphing
2 2
2 2 1x h y ka b
where c2 = a2 – b2 and c is the distance from the center to the foci.
Vertices are “a” units on the major axis and “b” units on the minor axis.
The foci are “c” units in the direction of the major axis.
![Page 8: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/8.jpg)
Ellipse – Table
2 2
2 2 1x h y ka b
Center: (h, k)Vertices: , ,h a k h k b
Foci: c2 = a2 – b2 a2 ≥ b2
,h c k
,h k cIf “a” is under “y”, the major axis is verticalIf “a” is under “x”, the major axis is horizontal
![Page 9: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/9.jpg)
Ellipse - Graphing
2 22 31
16 25x y
Graph:
Center: (2, -3)
Distance to vertices in x direction: 4
Distance to vertices in y direction: 5Distance to foci: c2=25-16 c2 = 9 c = 3
![Page 10: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/10.jpg)
Ellipse - Graphing
2 25 2 10 12 27 0x y x y Graph:
Complete the squares.2 25 10 2 12 27x x y y
2 25 2 ?? 2 6 ?? 27x x y y
2 25 2 1 2 6 9 27 5 18x x y y
2 25 1 2 3 50x y
2 21 31
10 25x y
-6y+9)=27+5(1)+2(9)
![Page 11: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/11.jpg)
Ellipse - Graphing
2 21 31
10 25x y
Graph:
Center: (-1, 3)
Distance to vertices in x direction:
Distance to vertices in y direction: Distance to foci: c2=|25 - 10| c2 = 15
c =
10
15
8
6
4
2
-2
-4
-5 5
5
![Page 12: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/12.jpg)
Ellipse – Find An EquationFind an equation of an ellipse with foci at (-1, -3) and (5, -3). The minor axis has a length of 4.
The center is the midpoint of the foci or (2, -3).
The minor axis has a length of 4 and the vertices must be 2 units from the center. Start writing the equation.
![Page 13: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/13.jpg)
Ellipse – Find An Equation
2 2
2 2 1x h y ka b
c2 = |a2 – b2|. Since the major axis is in the x direction, a2 > 169 = a2 – 16a2 = 25Replace a2 in the equation.
116)3()2( 2
2
2
yax
![Page 14: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/14.jpg)
Ellipse – Find An Equation
The equation is:
116)3(
25)2( 22
yx
![Page 15: Warm up Write the standard form of the equation: Then find the radius and the coordinates of the center. Graph the equation](https://reader036.vdocuments.site/reader036/viewer/2022062600/5a4d1b307f8b9ab05999aa5e/html5/thumbnails/15.jpg)
Practice
164)4(
100)3( 22
yx
Find the coordinates of the center, foci and vertices of the ellipse. Then graph