Definition of Ellipse

An ellipse is a locus of all points (x,y) such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant.

P

F1 F2

F1P + F2P = 2a

Major axisCenter (h,k)

Minor axis VertexVertex

Co-vertex

Co-vertex

focus

Center (h,k)

Major Axes:

Minor Axes:

Vertices:

Co-vertices:

Foci:There are TWO cases of an ellipse: Horizontal major axes and Vertical major axes

Horizontal Major Axis and C(0,0):

a2 > b2

a2 – b2 = c2

F1(–c, 0) F2 (c, 0)

y

x

V1(–a, 0) V2 (a, 0) (0, b)

(0, –b)

O

major axis = 2aminor axis = 2b

x2

a2

y2

b2+ = 1

a2 > b2

a2 – b2 = c2

(x – h)2

a2

(y – k)2

b2+ = 1

Horizontal Major Axis and C(h,k):

F1(0, –c)

F2 (0, c)

y

x

V1(0, –a)

V2 (0, a)

(b, 0)(–b, 0)

O

Vertical Major Axis and C(0,0):

a2 > b2

a2 – b2 = c2

x2

b2

y2

a2+ = 1

major axis = 2aminor axis = 2b

a2 > b2

a2 – b2 = c2

Vertical Major Axis and C(h,k):

(x – h)2

b2

(y – k)2

a2+ = 1

The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth. In the 17th century, Johannes Kepler discovered that each planet travels around the sun in an elliptical orbit

One of the reasons it was difficult to detect that orbits are elliptical is that the foci of the planetary orbits are relatively close to the center, making the ellipse nearly circular. To measure the ovalness of an ellipse, we use the concept of eccentricity.

DEFINITION:The eccentricity e of an ellipse is given by the ratio e = c/a

e e1

EX. 1: Write equations of ellipses graphed in the coordinate plane

EX. 2: Sketch the graph of each ellipse. Identify the center, the vertices, the co-vertices, and the foci for each ellipse.

EX.3: Find the coordinates of the center and vertices of an ellipse. Graph the ellipse.

center: (2, 1)

vertices:(–2, 1), (6, 1)

(x – 2)2

16

(y – 1)2

9+ = 1

EX. 4: Find the coordinates of the co-vertices, and foci of an ellipse. Graph the ellipse.

co-vertices:(2, 4), (2, –2)

foci:(2 – 7 , 1), (2 + 7 , 1)

(x – 2)2

16

(y – 1)2

9+ = 1

EX.5: Graph the ellipse.9x2 + 16y2 – 36x – 32y – 92 = 0

standard form:

(x – 2)2

16

(y – 1)2

9+ = 1

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