10.4 Ellipses p. 609

• An ellipse is a set of points such that the distance between that point and two fixed points called Foci remains constant f1 d1 d2 d3 f2 d4 d1 + d2 = d3 + d4

v 1 F 1 cv 1 c F 2 v 2 cv 2

• The line that goes through the Foci is the Major Axis.

• The midpoint of that segment between the foci is the Center of the ellipse (c) • The intersection of the major axis and the ellipse itself results in two points, the Vertices (v) • The line that passes through the center and is perpendicular to the major axis is called the Minor Axis • The intersection of the minor axis and the ellipse results in two points known as co-vertices

Example of ellipse with vertical major axis

Example of ellipse with horizontal major axis

Standard Form for Elliptical Equations Equation Major Axis (length is 2a) Minor Axis (length is 2b) Vertices Co Vertices

x

2

a

2 

y b

2 2  1 Horizontal Vertical (a,0) (-a,0) (0,b) (0,-b)

Note that a is the biggest number!!!

x

2

b

2 

y a

2 2  1 Vertical Horizontal (0,a) (0,-a) (b,0) (-b,0)

• The foci lie on the major axis at the points: • (c,0) (-c,0) for horizontal major axis • (0,c) (0,-c) for vertical major axis • Where c 2 = a 2 – b 2

Write the equation of an ellipse with center (0,0) that has a vertex at (0,7) & co-vertex at (-3,0) • Since the vertex is on the y axis (0,7) a=7 • The co-vertex is on the x-axis (-3,0) b=3 • The ellipse has a vertical major axis & is of the form

x

2

b

2 

y

2

a

2  1

x

9 2 

y

2 49  1

Given the equation 9x 2 + 16y 2 = 144 Identify: foci, vertices, & co-vertices • First put the equation in standard form: 9

x

2 144  16

y

2 144  144 144

x

2 16 

y

2 9  1

x

16 2 

y

2 9  1 • From this we know the major axis is

horizontal

& a=4, b=3 • • So the vertices are (4,0) & (-4,0) the co-vertices are (0,3) & (0,-3) • • To find the foci we use c 2 c 2 = a 2 = 16 – b – 9 2 • c = √7 • So the foci are at (√7,0) (-√7,0)

Assignment