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Section 9.2 The Hyperbola

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Overview In Section 9.1 we discussed the ellipse, one of four conic sections. Now we continue onto the hyperbola, which in itself is unusual because to obtain the graph you must intersect two cones instead of one.

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The Hyperbola A hyperbola is the set of all points in the plane the difference of who distances from two fixed points, called foci, is constant. The line through the foci intersects the hyperbola in two points, called vertices. The line segment that joins the vertices is called the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola.

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Pictures

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Case I: Center at the Origin There are two possible sub-cases when the hyperbola is centered at the origin: 1.The foci and vertices are on the x-axis (the transverse axis is horizontal). 2.The foci and vertices are on the y-axis (the transverse axis is vertical). In either case: a represents the distance from the center to a vertex. c represents the distance from the center to a focus.

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Case I (continued) The standard form of the equation of a hyperbola with center at the origin and horizontal transverse axis is:

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Case I (continued) The standard form of the equation of a hyperbola with center at the origin and vertical transverse axis is:

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Case I (continued) Important relationship:

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Case I (continued) Hyperbolas have asymptotes!! When the transverse axis is horizontal, the equations of the asymptotes are

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Asymptotes (continued) When the transverse axis is vertical, the equations of the asymptotes are

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Pictures

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Hyperbolas not centered at the origin The standard form of the equation of a hyperbola with horizontal transverse axis is:

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Continued The standard form of the equation of a hyperbola with vertical transverse axis is:

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Asymptotes For hyperbola with a horizontal transverse axis, the equations of the asymptotes are

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Asymptotes (continued) For hyperbola with a vertical transverse axis, the equations of the asymptotes are

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Pictures

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Examples Find the vertices, locate the foci, and give the equations of the asymptotes:

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More Examples—Draw The Picture! Write the equation of the hyperbola: 1.Foci at (0, -8) and (0, 8); vertices at (0, 1) and (0, -1) 2.Center (3, -2); focus (8, -2); vertex (7, -2)

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One More… Convert the equation to standard form by completing the square on x and y.

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