Transcript Section 8.2: Ellipses - Mrs. Fehling's Math Classes

Section 8.2: Ellipses
May 7, 2015

An ellipse is the set of all points in a plane whose
distances from 2 fixed points in the plane have a
constant sum.

The fixed points are foci of the ellipse.
The line through the foci is the focal axis.
The point midway between the foci is the center.
The points where the ellipse intersects its axis are
vertices.



Parts of an Ellipse
Major
axis
(2a)
b
a
c
a
Minor
axis
(2b)
b
a b c
2
2
2
Axes: Finding Relationships
An ellipse with a horizontal major axis has
an equation of
2
2
( x  h) ( y  k )

1
2
2
a
b
An ellipse with a vertical major axis has an
equation of
2
2
( y  k ) ( x  h)

1
2
2
a
b
Equation of an Ellipse
a)
x2 y 2

1
16 7
b) 9x2  4 y 2  36
c) ( y  3)2 ( x  7) 2

1
81
64
Find the center, vertices, and foci
of each ellipse.
a)
Foci: (0,3), (0,-3); major axis of length 10
b)
Major axis endpoints: (-2, -3), (-2, 7);
minor axis of length 4
c)
Foci: (1, -4), (5, -4); major axis
endpoints: (0, -4), (6, -4)
Find an equation in standard form
for each ellipse.
c
e
a
Prove that the graph of the equation is an ellipse.
Then, find vertices, foci, and eccentricity.
a)
9x  16 y  54 x  32 y  47  0
b)
4x  y  32x  16 y  124  0
2
2
2
2
Eccentricity of an Ellipse
( x  h) 2 ( y  k ) 2

 1,
 Given
2
2
a
b
x=h + a cos t
y = k + b sin t
( y  k ) ( x  h)
 Given

 1,
2
2
a
b
2
2
x=h + b cos t
y = k + a sin t
Parametric Equations of an Ellipse

( x  2)2 ( y  3) 2
Given

 1 , write the parametric equation.
16
9

( y  1) 2 ( x  7) 2

 1 , write the parametric equation.
Given
64
36
Practice w/ Parametric Equations
Application Problem
a)
x2 y 2

1
81 25
2
2
(
x

1)
(
y

3)
b)

1
2
4
Graph.