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