ellipse conic sections. ellipse the plane can intersect one nappe of the cone at an angle to the...

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