Embed Size (px)
Citation preview
10.5 Hyperbolas10.5 Hyperbolas
What you should learn:GoalGoal 11
GoalGoal 22
Graph and write equations of Hyperbolas.
Identify the Vertices and Foci of the hyperbola.
10.5 Hyperbolas10.5 Hyperbolas
GoalGoal 33 Identify the Foci and Asymptotes.
Hyperbolas
• Like an ellipse but instead of the sum of distances it is the difference
• A hyperbola is the set of all points P such that the differences from P to two fixed points, called foci, is constant
10.5 Hyperbolas10.5 Hyperbolas
Hyperbolas
• The line thru the foci intersects the hyperbola at two points (the vertices)
• The line segment joining the vertices is the transverse axis, and it’s midpoint is the center of the hyperbola.
• Has 2 branches and 2 asymptotes
• The asymptotes contain the diagonals of a rectangle centered at the hyperbolas center
10.5 Hyperbolas10.5 Hyperbolas
Standard Form of Hyperbola w/ center at origin
12
2
2
2
b
y
a
x
EquationTransverse
AxisAsymptotes Vertices
Horizontal y =+/- (b/a)x (+/-a,0)
Vertical y =+/- (a/b)x (0,+/-a)12
2
2
2
b
x
a
y
Foci lie on transverse axis, c units from the center c2 = a2+b2
10.5 Hyperbolas10.5 Hyperbolas
(0,b)
(0,-b)
Vertex (a,0)Vertex (-a,0)
Asymptotes
This is an example of a Horizontal Transverse axis (a, the biggest number, is under the x2 term
with the minus before the y)
FocusFocus
12
2
2
2
b
y
a
x
10.5 Hyperbolas10.5 Hyperbolas
Vertical Transverse axis
12
2
2
2
b
x
a
y
10.5 Hyperbolas10.5 Hyperbolas
3694 22 yx
149
22
yx
36 36 36
1
23 2
2
2
2
yx • a = 3 b = 2
• because term is positive, the transverse axis is horizontal & vertices are
(-3,0) & (3,0)
2x
Example) Graph the equation.
10.5 Hyperbolas10.5 Hyperbolas
Graph 4x2 – 9y2 = 36• Draw a rectangle centered at the origin.• Draw asymptotes.• Draw hyperbola.
Example)
1
23 2
2
2
2
yx
10.5 Hyperbolas10.5 Hyperbolas
10.5 Hyperbolas10.5 Hyperbolas
Write the equation of a hyperbola with foci (0,-3) & (0,3) and vertices (0,-2) & (0,2).
• Transverse axis is Vertical because foci & vertices lie on the y-axis
• Center is the origin because foci & vertices are equidistant from the origin
• Since c = 3 & a = 2, c2 = b2 + a2
• 9 = b2 + 4• 5 = b2
• +/-√5 = b 154
22
xy
10.5 Hyperbolas10.5 Hyperbolas
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How are the definitions of ellipse and hyperbola alike? How are they different?
assignmentassignment
Both involve all points a certain distance from 2 foci; For an ellipse, the sum of the distances is constant; for a hyperbola, the difference is constant.
10.5 Hyperbolas10.5 Hyperbolas
Page 618# 15 – 63 odd
