Transcript Conic Sections Hyperbolas Definition • The conic section formed by a plane which intersects both of the right conical surfaces • Formed when 0  

Conic Sections
Hyperbolas
Definition
• The conic section formed
by a plane which intersects
both of the right conical
surfaces
• Formed when 0    
or when the plane is
parallel to the axis
of the cone


Definition
• A hyperbola is the set of all points in the
plane where
 The difference between the distances
 From two fixed points (foci)
 Is a constant
PF2  PF1  k
k  2a
Geogebra
Demonstration
Experimenting with Definition
•
Turn on Explore geometric definition. A purple point will
appear on the hyperbola, along with two line segments
labeled L1 and L2. Drag the purple point around the
hyperbola.



•
Do the lengths of L1 and L2 change?
What do you notice about the absolute value of the differences of the
lengths?
How do these observations relate to the geometric definition of a
hyperbola.
Observe the values of L1, L2, and the difference | L1 − L2 |
as you vary the values of a and b.


How is the difference | L1 − L2 | related to the values of a and/or b?
(Hint: Think about multiples.)
Determine the difference | L1 − L2 | for a hyperbola where a = 3 and
b = 4. Use the Gizmo to check your answer.
Elements of An Ellipse
• Transverse axis
 Line joining the
intercepts
• Conjugate axis
 Passes through
center, perpendicular
to transverse axis
• Vertices
 Points where hyperbola intersects transverse
axis
Elements of An Ellipse
• Transverse Axis
 Length = 2a
• Foci
 Location (-c, 0), (c, 0)
• Asymptotes
 Experiment with Pythagorean relationship
Equations of An Ellipse
• Given equations of ellipse
2
x
y
 2 1
2
a b
 Centered at origin
 Opening right and left
 Equations of
asymptotes
b
y x
a
• Opening up and down
 Equations of
asymptotes
a
y x
b
2
b
y x
a
y 2 x2
 2 1
2
a b
a
y x
b
Try It Out
• Find
2
2
x
y

1
25 36
 The center
 Vertices
 Foci
 y  2
 Asymptotes
36
2
x  1


49
2
1
4x  25 y  16x  50 y 109  0
2
2
More Trials and Tribulations
• Find the equation in standard form of the
hyperbola that satisfies the stated conditions.
 Vertices at (0, 2) and (0, -2),
foci (0, 3) and (0, -3)
 Foci (1, -2) and (7, -2)
slope of an asymptote = 5/4
Assignment
• Hyperbola A
• Exercise set 6.3
• Exercises 1 – 25 odd
and 33 – 45 odd
Conic Sections
Eccentricity of a Hyperbola
• The hyperbola can be wide or narrow
Eccentricity of a Hyperbola
• As with eccentricity of
c
eccentricity 
an hyperbola, the formula is
a
 Note that for
hyperbolas c > a
 Thus eccentricity > 1
Try It Out
• If the vertices are (1, 6) and (1, 8) and the
eccentricity is 5/2
 Find the equation (standard form) of the
hyperbola
• The center of the hyperbola is at (-3, -3) and
the conjugate axis has length 6, and the
eccentricity = 2
 Find two possible hyperbola equations
Application – Locating Position
• For any point on a hyperbolic curve
 Difference between distances to foci is constant.
• Result: hyperbolas can be used to locate
enemy guns
“If the sound of an enemy gun is heard at two
listening posts and the difference in time is
calculated, then the gun is known to be located
on a particular hyperbola. A third listening post
will determine a second hyperbola, and then
the gun emplacement can be spotted as the
intersection of the two hyperbolas.”
Application – Locating Position
• The loran system navigator
 equipped with a map that gives curves, called loran lines
of position.
 Navigators find the time interval between these curves,
 Narrow down the area that their craft's position is in.
• Then switch to a different pair of loran transmitters
 Repeat the procedure
 Find another curve
representing the craft's
position.
Construction
• Consider a
the blue string
• Keep marker
against ruler
and with string
tight
• Keep end of
ruler on focus
F1 , string tied to other end
Graphing a Hyperbola on the TI
• As with the ellipse, the hyperbola is not a
function
• Possible to solve for y
 Get two expressions
 Graph each
• What happens
if it opens
right and left?
Graphing a Hyperbola on the TI
• Top and bottom of hyperbola branches are
graphed separately
 As with ellipses
you must
Ellipse with
Geogebra
Assignment
• Hyperbolas 2
• Exercise Set 6.3
• Exercises 27 – 31 odd
and 49 – 63 odd