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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. Graph - Example #1 Hyperbola

  14. Hyperbola - Graph Graph: Center: (-3, -2) The hyperbola opens in the “x” direction because “x” is positive. Transverse Axis: y = -2

  15. Hyperbola - Graph Graph: Vertices (2, -2) (-4, -2) Construct a rectangle by moving 4 unitsup and down from the vertices. Construct the diagonals of the rectangle.

  16. Hyperbola - Graph Graph: Draw the hyperbola touching the vertices and approaching the asymptotes. Where are the foci?

  17. Hyperbola - Graph Graph: The foci are 5 units from the center on the transverse axis. Foci: (-6, -2) (4, -2)

  18. Hyperbola - Graph Graph: Find the equation of the asymptote lines. 4 3 Use point-slope formy – y1 = m(x – x1) since the center is on both lines. -4 Slope = Asymptote Equations

  19. Graph - Example #2 Hyperbola

  20. Hyperbola - Graph Sketch the graph without a grapher: Recognition:How do you determine the type of conic section? Answer:The squared terms have opposite signs. Write the equation in hyperbolic form.

  21. Hyperbola - Graph Sketch the graph without a grapher:

  22. Hyperbola - Graph Sketch the graph without a grapher: Center: (-1, 2) Transverse Axis Direction: Up/Down Equation: x=-1 Vertices: Up/Down from the center or

  23. Hyperbola - Graph Sketch the graph without a grapher: Plot the rectangular points and draw the asymptotes. Sketch the hyperbola.

  24. Hyperbola - Graph Sketch the graph without a grapher: Plot the foci. Foci:

  25. Hyperbola - Graph Sketch the graph without a grapher: Equation of the asymptotes:

  26. Finding an Equation Hyperbola

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

  28. Conic Section Recogition

  29. 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”.

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