Section 8-3 The Hyperbola

Section 8-3 the geometric definition of a hyperbola standard form of a hyperbola with a center at (0, 0) translating a hyperbola – center at (h, k) graphing a hyperbola finding the equations of the asymptotes finding the equation of a hyperbola eccentricity and orbits reflective properties of hyperbola

Geometry of a Hyperbola hyperbola – the set of all points whose distance from two fixed points (the foci) have a constant difference all the points are coplanar the line through the foci is called the focal axis the midway point between the foci is called the center

Geometry of a Hyperbola F1F1 F2F2 V1V1 V2V2 center F 1 and F 2 are the foci V 1 and V 2 are the vertices (chord between called the transverse axis)

Geometry of a Hyperbola F1F1 F2F2 V1V1 V2V2 center d2d2 d1d1 F 1 and F 2 are the foci d 1 - d 2 = constant V 1 and V 2 are the vertices (chord between called the transverse axis)

Standard Form: Center (0, 0) 2a = length of the transverse axis (endpoints are the vertices) 2b = length of the conjugate axis c = focal radius (distance from the center to each foci) c 2 = a 2 + b 2 (use to find c)

Standard Form: Center (h, k)

Graphing a Hyperbola convert the equation into standard form, if necessary (complete the square) find and plot the center use “a” to plot the vertices (same direction as the variable a 2 is underneath) use “b” to plot two other points draw a rectangle using these four points draw the diagonals of the rectangle (dashed), these are the asymptotes draw in the hyperbola (use vertices) plot the foci using “c” (c is the distance from the center to each focus)

Equations of the Asymptotes the equations of the asymptotes can be found by replacing the 1 on the right-side of the equation with a 0 and then solving for y