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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
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