EllipsesEllipsesOn to Sec. 8.2a…On to Sec. 8.2a…

Geometry of an EllipseGeometry of an EllipseAn An ellipse ellipse is the set of all points in a plane whoseis the set of all points in a plane whosedistances from two fixed points in the plane have adistances from two fixed points in the plane have aconstant sum. The fixed points are the constant sum. The fixed points are the focifoci (plural (pluralof focus) of the ellipse. The line through the foci isof focus) of the ellipse. The line through the foci isthe the focal axisfocal axis. The point on the focal axis midway. The point on the focal axis midwaybetween the foci is the between the foci is the centercenter. The points where the. The points where theellipse intersect its axis are the ellipse intersect its axis are the verticesvertices of the ellipse. of the ellipse.

CenterFocus Focus

Vertex Vertex

Focal Axis

1F 2F

1d2d

P

1 2d dThe sum is a constant

Deriving the Equation of an EllipseDeriving the Equation of an Ellipse

1 ,0F c 2 ,0F c

b

a

,P x y

1 2 2PF PF a

Equation for the twodashed distances:

2 2 2 20 0 2x c y x c y a

Use the distance formula:

2 22 22x c y a x c y

22 2 2 2 2 2 2 22 4 4 2x cx c y a a x c y x cx c y

2 2 2b a c Something to notice:

Deriving the Equation of an EllipseDeriving the Equation of an Ellipse

2 2 2a x c y a cx

2 2 2 2 4 2 2 22 2a x cx c y a a cx c x

22 2 2 2 2 2 2 22 4 4 2x cx c y a a x c y x cx c y

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

2 2 2 2 2 2 2 4 2 2 22 2a x a cx a c a y a a cx c x

2 2

2 2 1x ya b

Deriving the Equation of an EllipseDeriving the Equation of an Ellipse2 2

2 2 1x ya b

This equation is the standard form of the equation of an ellipsecentered at the origin with the x-axis as its focal axis.

When the y-axis is the focal axis?2 2

2 2 1y xa b

Chord – segment with endpoints on the ellipseMajor Axis – chord lying on the focal axis

Minor Axis – chord through the center, perp. to the focal axis

The number a is the semimajor axis, and the number b is thesemiminor axis.

Ellipses with Center (0, 0)Ellipses with Center (0, 0)2 2

2 2 1x ya b

Standard Equation:

Focal Axis: x-axis

,0cFoci:

,0aVertices:

Semimajor Axis: a

Semiminor Axis: b

2 2 2a b c Pythagorean Relation:

ab

c

(0, b)

(0, –b)

(a, 0)(–a, 0) (–c, 0) (c, 0)

Ellipses with Center (0, 0)Ellipses with Center (0, 0)2 2

2 2 1y xa b

Standard Equation:

Focal Axis: y-axis

0, cFoci:

0, aVertices:

Semimajor Axis: a

Semiminor Axis: b

2 2 2a b c Pythagorean Relation:

a

b

c

(0, a)

(0, –a)

(b, 0)

(0, –c)

(0, c)

(–b, 0)

Some Practice Problems…Some Practice Problems…

Find the vertices and the foci of the ellipse2 24 9 36x y

First, write in standard form:

2 2

19 4x y

Because the larger number is under the “x” term, the focal axisis the x-axis. So… 2 9a 2 4b

2 2 2c a b 9 4 5

Vertices: 3,0 Foci: 5,0

Some Practice Problems…Some Practice Problems…Find an equation of the ellipse with foci (0, –3) and (0, 3) whoseminor axis has length 4. Sketch the ellipse and support yoursketch with a grapher.

The center is (0,0)

The foci are on the y-axis with c = 3

The semiminor axis is b = 4/2 = 2

Use the Pythagorean relationship to solve for a: 2 2 22 3a 13a

Some Practice Problems…Some Practice Problems…Find an equation of the ellipse with foci (0, –3) and (0, 3) whoseminor axis has length 4. Sketch the ellipse and support yoursketch with a grapher.

Standard Form:

13, 2, 3a b c

2 2

2 2 1y xa b

2 2

113 4y x

Can we graph by hand?

To check with a calculator, solve for y:

213 1 4y x

What happens when the center of an ellipse is not on the origin?

(h, k)

(h + c, k)(h – c, k)

(h – a, k) (h + a, k)

(h, k)

(h, k + a)

(h, k – a)

(h, k + c)

(h, k – c)

Ellipses with Center (h, k)Ellipses with Center (h, k)

• Standard Equation 2 2

2 2 1x h y ka b

• Focal Axisy k

• Foci ,h c k

• Vertices ,h a k

• Semimajor Axis a• Semiminor Axis b

• Pythagorean Relation 2 2 2a b c

Ellipses with Center (h, k)Ellipses with Center (h, k)

• Standard Equation 2 2

2 2 1y k x ha b

• Focal Axis x h

• Foci ,h k c

• Vertices ,h k a

• Semimajor Axis a• Semiminor Axis b

• Pythagorean Relation 2 2 2a b c

Guided PracticeFind the standard form of the equation for the ellipse whosemajor axis has endpoints (–2, –1) and (8, –1), and whose minoraxis has length 8.

How about starting with a diagram of the given info?

What is the general equation? 2 2

2 2 1x h y ka b

Where is the center midpoint of the major axis!!!

, 3, 1h k

Guided PracticeFind the standard form of the equation for the ellipse whosemajor axis has endpoints (–2, –1) and (8, –1), and whose minoraxis has length 8.

2 2

2 2 1x h y ka b

, 3, 1h k

Now, find the semimajor and semiminor axes:

8 42

b 8 2

52

a

Plug all of these values back into our general equation!!!

2 23 11

25 16x y

Whiteboard PracticeFind the center, vertices, and foci of the ellipse

2 22 51

9 49x y

Standard form: 2 25 2

149 9y x

Center:

Vertices: ,h k a 2,5 7

(–2, 5)(–2, 5)

Vertices:

(–2, 12), (–2, –2)(–2, 12), (–2, –2)

Foci: ,h k c 2,5 2 10 Foci: (–2, 11.325), (–2, –1.325)(–2, 11.325), (–2, –1.325)

Whiteboard PracticeFind an equation in standard form for the ellipse that satisfies thegiven conditions.Major axis endpoints are (–2, –3) and (–2, 7), minor axis length 4

Start with a diagram…

Center: 2,2Semimajor and semiminor axes: 5a 2b

Standard Form: 2 22 2

125 4y x

Whiteboard PracticeFind an equation in standard form for the ellipse that satisfies thegiven conditions.The foci are (–2, 1) and (–2, 5); the major axis endpoints are(–2, –1) and (–2, 7)

Start with a diagram…Center: 2,3 Semimajor axis:

2 2b a c

4a

Standard Form: 2 23 2

116 12y x

Semiminor axis: 2 24 2 12