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Ellipse
Conic Sections
Ellipse
The plane can
intersect one nappe of
the cone at an angle
to the axis resulting in
an ellipse.
Ellipse - Definition
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.
d1 + d2 = a constant value.
Finding An Equation
Ellipse
Ellipse - Equation
To find the equation of an ellipse, let the center be at
(0, 0). The vertices on the axes are at (a, 0), (-a, 0),
(0, b) and (0, -b). The foci are at (c, 0) and (-c, 0).
Ellipse - Equation
According to the definition. The sum of the distances
from the foci to any point on the ellipse is a constant.
Ellipse - Equation
The distance from the foci to the point (a, 0) is 2a.
Why?
Ellipse - Equation
The distance from (c, 0) to (a, 0) is the same as from
(-a, 0) to (-c, 0).
Ellipse - Equation
The distance from (-c, 0) to (a, 0) added to the
distance from (-a, 0) to (-c, 0) is the same as going
from (-a, 0) to (a, 0) which is a distance of 2a.
Ellipse - Equation
Therefore, d1 + d2 = 2a. Using the distance formula,
( x  c ) 2  y 2  ( x  c ) 2  y 2  2a
Ellipse - Equation
Simplify:
( x  c ) 2  y 2  ( x  c ) 2  y 2  2a
( x  c )  y  2a  ( x  c )  y
2
2
2
2
Square both sides.
( x  c ) 2  y 2  4a 2  4a ( x  c ) 2  y 2  ( x  c ) 2  y 2
Subtract y2 and square binomials.
x 2  2 xc  c 2  4a 2  4a ( x  c) 2  y 2  x 2  2 xc  c 2
Ellipse - Equation
Simplify:
x 2  2 xc  c 2  4a 2  4a ( x  c) 2  y 2  x 2  2 xc  c 2
Solve for the term with the square root.
4 xc  4a 2  4a ( x  c) 2  y 2
xc  a 2  a ( x  c) 2  y 2
Square both sides.
 xc  a

2 2

 a ( x  c)2  y 2

2
Ellipse - Equation
Simplify:
 xc  a 
2 2

 a ( x  c)  y
2
2

2
x c  2 xca  a  a  x  2 xc  c  y 
x 2c 2  2 xca 2  a 4  a 2 x 2  2 xca 2  a 2c 2  a 2 y 2
2 2
2
4
2
2
2
x c a  a x a c a y
2 2
4
2 2
2 2
2
2
Get x terms, y terms, and other terms together.
2 2
2 2
2 2
2 2
4
x
c

a
x

a
y

a
c

a


2
Ellipse - Equation
Simplify:
x c
2 2
a x a y  a c a
2
2
2
2
2 2
4
2
2
2
2 2
2
2
2
c

a
x

a
y

a
c

a




Divide both sides by a2(c2-a2)
c
2
 a2  x2
2
2
a2  c2  a2 
a y
 2 2
 2 2
2
2
2
2
a c  a  a c  a  a c  a2 
x2
y2
 2
1
2
2
a c  a 
Ellipse - Equation
2
2
x
y
 2
1
2
2
a c  a 
Change the sign and run the negative through the
denominator.
2
2
x
y
 2 2 1
2
a a  c 
At this point, let’s pause and investigate a2 – c2.
Ellipse - Equation
d1 + d2 must equal 2a. However, the triangle created
is an isosceles triangle and d1 = d2. Therefore, d1 and
d2 for the point (0, b) must both equal “a”.
Ellipse - Equation
This creates a right triangle with hypotenuse of length
“a” and legs of length “b” and “c”. Using the
pythagorean theorem, b2 + c2 = a2.
Ellipse - Equation
We now know…..
x2
y2
 2 2 1
2
a a  c 
and b2 + c2 = a2
b2 = a2 – c2
Substituting for a2 - c2
2
2
x
y
 2 1
2
a
b
where c2 = |a2 – b2|
Ellipse - Equation
The equation of an ellipse centered at (0, 0) is ….
2
2
where a2 = b2 + c2 and
c is the distance from the
center to the foci.
x
y
 2 1
2
a
b
Shifting the graph over h units and up k units, the
center is at (h, k) and the equation is
 x  h
a
2
2
y k


b
2
2
1
where a2 = b2 + c2 and
c is the distance from the
center to the foci.
Ellipse - Graphing
 x  h
a
2
2
y k


b
2
where a2 = b2 + c2 and
c is the distance from the
center to the foci.
2
1
Vertices are “a” units in
the x direction an “b”
units in the y direction.
b
a
c
a
c
b
The foci are “c” units in
the direction of the
longer (major) axis.
Graph - Example #1
Ellipse
Ellipse - Graphing
Ellipse - Graphing
Graph:
 x  2
16
2
y  3


25
2
1
Graph - Example #2
Ellipse
Ellipse - Graphing
Graph:
2
5 x  2 y  10 x  12 y  27  0
2
Find An Equation
Ellipse
Ellipse – Find An Equation
Find an equation of an ellipse with
foci at (-1, -3) and (5, -3). The
minor axis has a length of 4.
Ellipse – Story Problem
A semielliptical arch is to have a span of 100 feet.
The height of the arch, at a distance 40 feet from
the center is to be 100 feet. Find the height of the
arch at its center.
Ellipse – Story Problem
A hall 100 feet in length is to be designed into a
whispering gallery. If the foci are located 25 feet
from the center, how high will the ceiling be at the
center?
Assignment:
Wksheet #4-7**, 20-23, 33, 38, 46, 47
**Graph and find center, major vertices,
minor vertices, and foci

Please use graph paper!!