Transcript 10.5 Hyperbolas p.615 Hyperbolas • Like an ellipse but instead of the sum of distances it is the difference • A hyperbola is the.

10.5 Hyperbolas
p.615
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
• The line thru the foci intersects the hyperbola @
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
Asymptotes
Vertex (-a,0)
(0,b)
Vertex (a,0)
Focus
Focus
(0,-b)
This is an example of a horizontal transverse axis
(a, the biggest number, is under the x2 term
with the minus before the y)
Vertical transverse axis
2
2
y
x
 2 1
2
a
b
Standard Form of Hyperbola w/
center @ origin
Transvers
Equation
Asymptotes Vertices
e Axis
x2 y2
 2  1 Horizontal
2
a
b
y=+/- (b/a)x
(+/-a,o)
y2 x2
 2  1 Vertical
2
a
b
y=+/- (a/b)x
(0,+/-a)
Foci lie on transverse axis, c units from the center c2 = a2+b2
Graph 4x2 – 9y2 = 36
• Write in standard form (divide through by 36)
• a=3 b=2 – because x2 term is ‘+’ transverse
axis is horizontal & vertices are (-3,0) & (3,0)
• Draw a rectangle centered at the origin.
• Draw asymptotes.
• Draw hyperbola.
Write the equation of a hyperbole
with foci (0,-3) & (0,3) and vertices
(0,-2) & (0,2).
• Vertical because foci & vertices lie on the y-axis
• Center @ origin because f & v are equidistant from
the origin
• Since c=3 & a=2, c2 = b2 + a2
2
2
2
•
9=b +4
•
5 = b2
•
+/-√5 = b
y
x

1
4
5
Assignment