Ellipse

Conic Sections

EllipseThe plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse.

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.

Finding An Equation

Ellipse

Ellipse - EquationTo 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 - EquationAccording to the definition. The sum of the distances from the foci to any point on the ellipse is a constant.

Ellipse - EquationThe distance from the foci to the point (a, 0) is 2a. Why?

Ellipse - EquationThe distance from (c, 0) to (a, 0) is the same as from (-a, 0) to (-c, 0).

Ellipse - EquationThe 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 - EquationTherefore, d1 + d2 = 2a. Using the distance formula,

2 2 2 2( ) ( ) 2x c y x c y a

Ellipse - EquationSimplify:

2 2 2 2( ) ( ) 2x c y x c y a

2 2 2 2( ) 2 ( )x c y a x c y

Square both sides.

2 2 2 2 2 2 2( ) 4 4 ( ) ( )x c y a a x c y x c y Subtract y2 and square binomials.

2 2 2 2 2 2 22 4 4 ( ) 2x xc c a a x c y x xc c

Ellipse - EquationSimplify:

2 2 2 2 2 2 22 4 4 ( ) 2x xc c a a x c y x xc c Solve for the term with the square root.

2 2 24 4 4 ( )xc a a x c y

2 2 2( )xc a a x c y Square both sides.

222 2 2( )xc a a x c y

Ellipse - EquationSimplify:

222 2 2( )xc a a x c y

2 2 2 4 2 2 2 22 2x c xca a a x xc c y 2 2 2 4 2 2 2 2 2 2 22 2x c xca a a x xca a c a y

2 2 4 2 2 2 2 2 2x c a a x a c a y Get x terms, y terms, and other terms together.

2 2 2 2 2 2 2 2 4x c a x a y a c a

Ellipse - EquationSimplify:

2 2 2 2 2 2 2 2 4x c a x a y a c a

2 2 2 2 2 2 2 2c a x a y a c a

Divide both sides by a2(c2-a2)

2 2 2 2 2 22 2

2 2 2 2 2 2 2 2 2

c a x a c aa y

a c a a c a a c a

2 2

2 2 21

x y

a c a

Ellipse - Equation

At this point, let’s pause and investigate a2 – c2.

2 2

2 2 21

x y

a c a

Change the sign and run the negative through the denominator.

2 2

2 2 21

x y

a a c

Ellipse - Equationd1 + 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 - EquationThis creates a right triangle with hypotenuse of length “a” and legs of length “b” and “c”. Using the pythagorean theorem, b2 + c2 = a2.

Ellipse - EquationWe now know…..

2 2

2 2 21

x y

a a c

and b2 + c2 = a2

b2 = a2 – c2

Substituting for a2 - c2

2 2

2 21

x y

a b where c2 = |a2 – b2|

Ellipse - Equation

2 2

2 21

x h y k

a b

The equation of an ellipse centered at (0, 0) is ….

2 2

2 21

x y

a 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.

Ellipse - Graphing 2 2

2 21

x h y k

a b

where c2 = |a2 – b2| andc is the distance from the center to the foci.

Vertices are “a” units in the x direction and “b” units in the y direction.

aa

b

b The foci are “c” units in the direction of the longer (major) axis.

cc

Graph - Example #1

Ellipse

Ellipse - Graphing

2 22 3

116 25

x y

Graph:

Center: (2, -3)

Distance to vertices in x direction: 4

Distance to vertices in y direction: 5

Distance to foci: c2=|16 - 25| c2 = 9 c = 3

Ellipse - Graphing

2 22 3

116 25

x y

Graph:

Center: (2, -3)

Distance to vertices in x direction: 4

Distance to vertices in y direction: 5

Distance to foci: c2=|16 - 25| c2 = 9 c = 3

Graph - Example #2

Ellipse

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 3

110 25

x y

Ellipse - Graphing

2 21 3

110 25

x 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

Find An Equation

Ellipse

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.

Ellipse – Find An Equation

2 2

2 21

x h y k

a b

2 2

2

2 31

4

x y

a

c2 = |a2 – b2|. Since the major axis is in the x direction, a2 > 49 = a2 – 4

a2 = 13Replace a2 in the equation.

Ellipse – Find An Equation

2 22 3

113 4

x y

The equation is:

Ellipse – Table 2 2

2 21

x h y k

a b

Center: (h, k)

Vertices: , ,h a k h k b

Foci: c2 = |a2 – b2|

If a2 > b2 ,h c k

If b2 > a2 ,h k c