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Section 9-5 Hyperbolas

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Objectives I can write equations for hyperbolas I can graph hyperbolas I can Complete the Square to obtain Standard Format of the equation

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Hyperbola Definition A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to the foci is a constant.

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Basic Diagram Transverse Axis F1F2 Conjugate Axis Vertex

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Equation Information Transverse Axis = 2a units Conjugate Axis = 2b units Vertex is a units from the center point (h, k) Focus Point is c units from center point

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Equations For Hyperbolas with Foci at (-c, 0) and (c, 0) and center point (h, k) opening left and right (horizontal transverse axis) For Hyperbolas with Foci at (0, -c) and (0, c) opening up and down (vertical transverse axis) Where c 2 = a 2 + b 2

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Hyperbolas

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Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 1 Write the equation for the hyperbola with transverse axis length 8 units and foci at (6,0) and (-6,0) Based on foci the transverse axis is horizontal Since length = 2a = 8; then a = 4 Then c 2 = a 2 + b 2 6 2 = 4 2 + b 2 36 = 16 + b 2 b 2 = 20

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Draw the graph: a 2 = 16, so a = 4 b 2 = 25, so b = 5 c 2 = a 2 + b 2 c 2 = 16 + 25 = 41 c = 6.4 Center Point (-2, 5) Transverse axis is horizontal = 2a = 8 units Asymptotes y = +/- 5/4 x

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EXAMPLE 1 Graph an equation of a hyperbola Graph 25y 2 – 4x 2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola. SOLUTION STEP 1 Rewrite the equation in standard form. 25y 2 – 4x 2 = 100 Write original equation. 25y 2 100 – 4x 2 100 = Divide each side by 100. y 2 4 – y 2 25 = 1 Simplify.

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Complete the square 4y 2 - 2x 2 + 16y – 4x - 10 = 0 (4y 2 + 16y) +(-2x 2 - 4x) = 10 4(y 2 + 4y) -2(x 2 + 2x) = 10 4(y 2 + 4y + 4) -2(x 2 + 2x + 1) = 10 + 16 – 2 4(y + 2) 2 – 2(x + 1) 2 = 24 Now divide all terms by 24

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Homework Worksheet 10-8

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