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Hyperbolas Sec. 8.3a

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Definition: Hyperbola A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola. How is this different from an ellipse???

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Definition: Hyperbola Center Vertex Focus Focal Axis

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Deriving the Equation of a Hyperbola 0 Notice: Combining: Distance Formula:

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Deriving the Equation of a Hyperbola

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Let Divide both sides by

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Deriving the Equation of a Hyperbola This equation is the standard form of the equation of a hyperbola centered at the origin with the x-axis as its focal axis. When the y-axis is the focal axis? Chord – segment with endpoints on the hyperbola Transverse Axis – chord lying on the focal axis, connecting the vertices (length = 2a) Conjugate Axis – segment (length = 2b) that is perp. to the focal axis and has the center of the hyperbola as its midpoint

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This equation is the standard form of the equation of a hyperbola centered at the origin with the x-axis as its focal axis. When the y-axis is the focal axis? Semitransverse Axis – the number “a” Semiconjugate Axis – the number “b”

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Deriving the Equation of a Hyperbola The hyperbola has two asymptotes, which can be found by replacing the “1” in the equation with a “0”: Solve for y Drawing Practice: Steps to sketching the hyperbola

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Hyperbolas with Center (0, 0) Standard Equation Focal Axis x-axis Foci Vertices Semitrans. Axis Semiconj. Axis Pythagorean Relation Asymptotes y-axis

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Hyperbolas with Center (0, 0)

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Guided Practice Find the vertices and the foci of the hyperbola Standard Equation: Sketch the hyperbola? Vertices:Foci:

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Guided Practice Find an equation of the hyperbola with foci (0, –3) and (0, 3) whose conjugate axis has length 4. Sketch the hyperbola and its asymptotes, and support your sketch with a grapher. General Equation: c = 3b = 2 a = 5 Standard Equation: The Sketch???

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Let’s see some hyperbolas whose centers are not on the origin…

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Hyperbolas with Center (h, k) Standard Equation Focal Axis Foci Vertices Semitransverse Axis Semiconjugate Axis Pythagorean Relation Asymptotes

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Hyperbolas with Center (h, k) Standard Equation Focal Axis Foci Vertices Semitransverse Axis Semiconjugate Axis Pythagorean Relation Asymptotes

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Guided Practice Find the standard form of the equation for the hyperbola whose transverse axis has endpoints (–2, –1) and (8, –1), and whose conjugate axis has length 8. Start with a diagram? General Equation: The center is the midpoint of the transverse axis:

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Find the standard form of the equation for the hyperbola whose transverse axis has endpoints (–2, –1) and (8, –1), and whose conjugate axis has length 8. Semitransverse Axis: Semiconjugate Axis: Specific Equation:

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Guided Practice Find the center, vertices, and foci of the given hyperbola. Center: Vertices: The graph? Foci:

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Guided Practice Find an equation in standard form for the hyperbola with transverse axis endpoints (–2, –2) and (–2, 7), slope of one asymptote 4/3. Start with a graph? Center is the midpoint of the transverse axis: Find a, the semi-transverse axis: Asymptote slope is a/b:

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Guided Practice Find an equation in standard form for the hyperbola with transverse axis endpoints (–2, –2) and (–2, 7), slope of one asymptote 4/3. General equation: Plug in data:

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