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Hyberbola

Hyberbola. Conic Sections. The plane can intersect two nappes of the cone resulting in a hyperbola. 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.

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Hyberbola

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  1. Hyberbola Conic Sections

  2. The plane can intersect two nappes of the cone resulting in a hyperbola. Hyperbola

  3. 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.

  4. Finding An Equation Hyperbola

  5. 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

  6. Hyperbola - Equation Find the equation by setting the difference in the distance from the two foci equal to 2a. | d1 – d2 | = 2a

  7. Hyperbola - Equation Simplify: Remove the absolute value by using + or -. Get one square root by itself and square both sides.

  8. Hyperbola - Equation Subtract y2 and square the binomials. Solve for the square root and square both sides.

  9. Hyperbola - Equation Square the binomials and simplify. Get x’s and y’s together on one side.

  10. Hyperbola - Equation Factor. Divide both sides by a2(c2 – a2)

  11. 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:

  12. 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 (+).

  13. Example #1 Hyperbola

  14. Find the center, vertices, foci, and asymptotes of the hyperbola. Then graph.

  15. Example #2 Hyperbola

  16. Find the standard form of the equation for the hyperbola. Then graph and find center, vertices, foci and asymptotes

  17. Finding an Equation Hyperbola

  18. Hyperbola – Find an Equation Find the equation of a hyperbola with foci at (2, 6) and (2, -4). The transverse axis length is 6.

  19. Finding an Equation Hyperbola

  20. Hyperbola – Find an Equation One focus is at

  21. Finding an EquationA problem for CSI! Hyperbola

  22. 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.

  23. 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. Let the vertices be at (+z, 0). Assuming z>0, then(z-(-1000)) – (1000-z) = 594z+1000-1000+z = 5942z = 594 or z = 297.

  24. 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.

  25. Hyperbola – Find an Equation V(294, 0) V(294, 0) V V The model is:

  26. Hyperbola – Find an Equation The gunshot was calculated to be at some point along the hyperbola.

  27. Conic Section Recogition

  28. 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”.

  29. Assignment: (Please do your assignment on graph paper!!) Finish Wksheet #26, 27, 34, 39, 48, 49, and p. 803 #68 (Yesterday’s Assign):Wksheet #8-11**, 24, 25, 39 **Graph and find center, vertices, foci, and asymptotes.

  30. Story Problems 36p - 72 square units 1. Find the area of the shaded region assuming the quadrilateral inside the circle is a square. The equation of the circle is x2 + y2 = 36.

  31. Story Problem: 4.17 ft along the base, on the axis of symmetry 2. A mirror is shaped like a paraboloid of revolution and will be used to concentrate the rays of the sun at its focus, creating a heat source. If the mirror is 20 feet across at its opening and is 6 feet deep, where will the heat source be concentrated?

  32. Story Problem 73.69 feet 3. An arch for a bridge over a highway is in the form of half an ellipse. The top of the arch is 20 feet above the ground level. The highway has four lanes, each 12 feet wide; a center safety strip 8 feet wide; and two side strips, each four feet wide. What should the span of the bridge be if the height 28 feet from the center is to be 13 feet?

  33. Story Problem: 5236 feet North of point A Suppose two people standing 1 mile (5280 feet) apart both see a flash of lightning. After a period of time, the first person standing at point A hears the thunder. Two seconds later, the second person standing at point B hears the thunder. If the person standing at point B is due west of the person standing at point A and if the lightning strike is known to occur due north of the person standing at point A, where did the lightning strike? (Note: speed of sound is 1100 ft/sec)

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