Today in Precalculus Turn in graded worksheet Notes: Conic Sections - Hyperbolas Homework.

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Presentation transcript:

Today in Precalculus Turn in graded worksheet Notes: Conic Sections - Hyperbolas Homework

Hyperbolas Definition: A hyperbola is the set of all points in a plane whose distances from two fixed points in a plane have a constant difference. The fixed points are the foci (F). The line through the foci is the focal axis. The point on the focal axis midway between the two foci is the center (C). The points on the hyperbola that intersect with the focal axis are the vertices (V). FF C focal axis VV

Standard form for the equation of a hyperbola centered at the origin with the x-axis as its focal axis is There is a pythagorean relationship between a,b, and c: c 2 = a 2 + b 2 F(-c,0)F (c,0) C(0,0) (-a,0)(a,0)

A line segment with endpoints on an hyperbola is a chord of the hyperbola The chord lying on the focal axis connecting the vertices is the transverse axis of the hyperbola and has a length of 2a. The value for a is the semitransverse axis. The segment through the center perpendicular to the focal axis is the conjugate axis of the hyperbola and has a length of 2b. The value of b is the semiconjugate axis The hyperbola also has two slant asymptotes whose equations depend on a and b.

A hyperbola centered at the origin with the y-axis as its focal axis has the form:

Hyperbola with center (0,0) Standard Equation Focal axisx-axisy-axis Foci(±c, 0)(0, ±c) Vertices(±a, 0)(0, ±a) Semitransverse axisaa Semiconjugate axisbb Pythagorean relationc 2 =a 2 + b 2 Asymptotes

Example 1 Find the vertices and foci of the hyperbola 4x 2 – 9y 2 = 36 Vertices: (-3, 0), (3, 0) c 2 = = 13 c = ±3.6 Foci: (-3.6, 0), (3.6,0)

Example 2 Find an equation of the hyperbola with foci (0, ) and transverse axis length From the foci: c = and focal axis is y-axis 2a = a = 17 = 13 + b 2 4 = b 2

Sketching hyperbolas Vertices (0, -3.6), (0, 3.6) Points (-2, 0), (2,0)

Graphing a hyperbola Like the other conic sections, must solve the equation for y

Homework Pg 663: 1, 2, 5, 6, 11-14, 23-26