Lesson 9.3 Hyperbolas

Hyperbola Set of all points where the difference between the distances to two fixed points (foci) is a positive constant. 20 cm 12 cm 7cm 15 cm 12 cm 4cm 10cm 2cm Focus Focus

Other parts of a hyperbola: Transversal axis can run vertically Axis (transverse) Center Vertices Foci

Equation of a Hyperbola Similar to ellipse Similarities/Differences: Subtraction between x2 and y2 terms Variable above a determines the direction of axis a is still the distance from center to a vertex c is still the distance from center to a focus c is now larger so… Horizontal axis Vertical axis

Example Find the standard form of the equation with foci (-1, 2) and (5, 2) and vertices (0, 2) and (4, 2).

Asymptotes Question: What is b in an hyperbola? It is still a distance from the center in the opposite direction of the axis How it applies to a parabola has to do with a new part unique to hyperbolas. Asymptotes Lines that bound the hyperbola Pass through the diagonals of a rectangle with dimensions 2a and 2b

Asymptotes (h, k + b) (h - a, k) (h, k) (h + a, k) (h, k - b) Conjugate axis (h, k + b) (h - a, k) (h, k) (h + a, k) (h, k - b) b is the distance from center to edge of rectangle along conjugate axis

Equations for Asymptotes Horizontal hyperbola (transverse axis) Vertical hyperbola (transverse axis) b is up/down – a is left/right a is up/down – b is left/right slope is rise over run → slope is rise over run →

Example Sketch the graph of 4x2 – y2 = 16, include the asymptotes.

Example Find the standard form of the hyperbola with vertices (3, -5), (3, 1) and asymptotes

Eccentricity of an Hyperbola where e > 1 (since c is larger than a) large e → flatter curve e close to 1 → more curved - pointed Problem Set 9.3