Ellipses & Hyperbolas

Advanced GeometryConic Sections

Lesson 4

Definition – the set of all points in a plane that the sum of the distances from two given points, called the foci, is constant

C

Major Axis

Minor Axis

VF

Ellipses

FV

V

V

Equation (a² > b²)

Center

Foci

equation

vertices

equation

vertices

2 2

2 21

x h y k

a b

2 2

2 2 1y k x h

a b

Major Axis

Minor Axis

( , )h k ( , )h k

( , )h c k ( , )h k c

y k x h( , )h a k ( , )h k ax h y k

( , )h k b ( , )h b k

2 2c a b

Example:For the equation of each ellipse or hyperbola, find all information listed. Then graph.

2 21 2

136 9

x y

Center:

Foci:

Length of the major axis:

Length of the minor axis:

Hyperbola

C

F

Transverse Axis

Conjugate AxisV

Asymptote

V

F

AsymptoteDefinition – the set of all points in a plane that the absolute value of the distance from two given points in the plane, called the foci, is constant

Equation of a Hyperbola

Center

Foci

Vertices

Slopes of the Asymptotes

Direction of Opening

2 2

2 2 1x h y k

a b

2 2

2 2 1y k x h

a b

( , )h k ( , )h k

( , )h c k ( , )h k c( , )h a k ( , )h k a

2 2c a b

b

a

a

b

left and right upand down

2 21 1

125 16

y x

Center:

Vertices:

Foci:

Slopes of the asymptotes:

Example:For the equation of each ellipse or hyperbola, find all information listed. Then graph.

Example:Using the graph below, write the equation for the ellipse or hyperbola.

Example:Using the graph below, write the equation for the ellipse or hyperbola.

Example:Write the equation of the ellipse or hyperbola that meets each set of conditions.

The foci of an ellipse are (-5, 3) and (3, 3) and the minor axis is 6 units long.

Example:Write the equation of the ellipse or hyperbola that meets each set of conditions.

The vertices of a hyperbola are (0,-3) and (0, -8) and the length of the conjugate axis is units long.2 6

Example:Write each equation in standard form. Determine if it is an ellipse or a hyperbola.

2 232 1 18 4 144 0x y