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Hyberbola Conic Sections Hyperbola The plane can intersect
twonappes of the cone resultingin a hyperbola. Hyperbola -
Definition
A hyperbola is the set of all points in a plane such that the
difference in the distances from two points (foci) is constant. |
d1 d2 | is a constant value. Finding An Equation Hyperbola
Hyperbola - Definition
What is the constant value for the difference in the distance from
the two foci?Let the two foci be (c, 0) and (-c, 0).The vertices
are (a, 0) and (-a, 0). | d1 d2 | is the constant. If the length of
d2 is subtracted from the left side of d1, what is the length which
remains? | d1 d2 | = 2a Hyperbola - Equation Find the equation by
setting the difference in the distance from the two foci equal to
2a. | d1 d2 | = 2a Hyperbola - Equation Simplify:
Remove the absolute value by using + or -. Get one square root by
itself and square both sides. Hyperbola - Equation Subtract y2 and
square the binomials.
Solve for the square root and square both sides. Hyperbola -
Equation Square the binomials and simplify.
Get xs and ys together on one side. Hyperbola - Equation Factor.
Divide both sides by a2(c2 a2) Hyperbola - Equation Let b2 = c2 a2
where c2 = a2 + b2
If the graph is shifted over h units and up k units, the equation
of the hyperbola is: Hyperbola - Equation where c2 = a2 + b2
Recognition: How do you tell a hyperbola from an ellipse? Answer: A
hyperbola has a minus (-) between the terms while an ellipse has a
plus (+). Example #1 Hyperbola Find the center, vertices, foci, and
asymptotes of the hyperbola
Find the center, vertices, foci, and asymptotes of the
hyperbola.Then graph. Example #2 Hyperbola Find the center,
vertices, foci, and asymptotes of the hyperbola
Find the center, vertices, foci, and asymptotes of the
hyperbola.Then graph. Finding an Equation Example 3
Hyperbola Hyperbola Find an Equation
Find the equation of a hyperbola with foci at (5, 0) and (-5, 0)
and transverse axis length of 8. Finding an Equation Example
4
Hyperbola Hyperbola Find an Equation
Find the equation of a hyperbola with center at the origin,
perimeter of central rectangle is 24 units, and vertices are at (0,
2) and (0, -2). Finding an Equation - Challenge A problem for
CSI!
Hyperbola Hyperbola Find an Equation
The sound of a gunshot was recorded at one microphone 0.54 seconds
before being recorded at a second microphone.If the two microphones
are 2,000 ft apart. Provide a model for the possible locations of
the gunshot.(The speed of sound is 1100 ft/sec.) The time between
the shots can be used to calculate the difference in the distance
from the two microphones. 1100 ft/sec * 0.54 sec = 594 ft. The
constant difference in distance from the microphones is 594 ft.
Since the difference is constant, the equation must be a
hyperbola.The points on the hyperbola are possible positions for
the gunshot. Hyperbola Find an Equation
Two microphones are stationed 2,000 ft apart.The difference in
distance between the microphones is 594 ft. Let the center be at
(0,0).The foci must be 2,000 ft apart. V The vertices are a
possible position for the gunshot.The difference in the distance
must be 594 feet between the vertices. Hyperbola Find an
Equation
V(-2970, 0) V(297, 0) V V Oops!We could have remembered the
constant difference in distance is 2a!2a = 594, a = 297. Start
finding the model of the hyperbola. 2972 = 88209 The distance from
the center to the foci (c) is 1000 ft.Find b. Hyperbola Find an
Equation
V(294, 0) V(294, 0) V V The model is: Hyperbola Find an
Equation
The gunshot was calculated to be at some point along the hyperbola.
Conic Section Recogition Recognizing a Conic Section
Parabola - One squared term.Solve for the term which is not
squared.Complete the square on the squared term. Ellipse - Two
squared terms.Both terms are the same sign. Circle - Two squared
terms with the same coefficient. Hyperbola - Two squared terms with
opposite signs. 10.5 Hyperbola- Assignment