Warm up

•Write the standard form of the equation:

• Then find the radius and the coordinates of the center.

• Graph the equation

0132822 yxyx

Lesson 10-3 EllipsesObjective: To use and determine the standard and general forms of the equation of an ellipseTo graph ellipses

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. The center of the ellipse is the midpoint of the

line segment that joins the foci

An ellipse has 2 axes of symmetry. The longer one contains the foci and is the major axis. The shorter one is called the minor axis.

F1: (-c,0) F2: (c,0)

Minor Axis

Major Axis

Vertex Vertex

Vertex

Vertex

Ellipse - Equation

2 2

2 2 1x h y ka b

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

2 2

2 2 1x ya 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.

Equation of an Ellipse: Center at (0,0); Foci at (0, c) and (0, -c); Major Axis is Vertical

12

2

2

2

ay

bx where a2 ≥ b2 and

b2 = a2 - c2

The major axis is the y - axis. The vertices are at (0, -a) and (0, a)

aab

b cc

Ellipse - Graphing

2 2

2 2 1x h y ka b

where c2 = a2 – b2 and c is the distance from the center to the foci.

Vertices are “a” units on the major axis and “b” units on the minor axis.

The foci are “c” units in the direction of the major axis.

Ellipse – Table

2 2

2 2 1x h y ka b

Center: (h, k)Vertices: , ,h a k h k b

Foci: c2 = a2 – b2 a2 ≥ b2

,h c k

,h k cIf “a” is under “y”, the major axis is verticalIf “a” is under “x”, the major axis is horizontal

Ellipse - Graphing

2 22 31

16 25x y

Graph:

Center: (2, -3)

Distance to vertices in x direction: 4

Distance to vertices in y direction: 5Distance to foci: c2=25-16 c2 = 9 c = 3

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 31

10 25x y

-6y+9)=27+5(1)+2(9)

Ellipse - Graphing

2 21 31

10 25x 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

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 2 1x h y ka b

c2 = |a2 – b2|. Since the major axis is in the x direction, a2 > 169 = a2 – 16a2 = 25Replace a2 in the equation.

116)3()2( 2

2

2

yax

Ellipse – Find An Equation

The equation is:

116)3(

25)2( 22

yx

Practice

164)4(

100)3( 22

yx

Find the coordinates of the center, foci and vertices of the ellipse. Then graph