10.4Hyperbolas

JMerrill 2010

DefinitionA hyperbola is the set of all points in a plane, the difference of whose distances from two distinct fixed point (foci) is a positive constant.

Equations of Hyperbolas

( )b

y k x ha

( )a

y k x hb

Writing the EquationFind the equation of the

hyperbola with vertices(-3, 2), (3, 2) and foci (-5, 2), (5, 2). Graph.  

22 31

9 16

yx

Find and Graph the HyperbolaState the direction of the transverse axis,

sketch a graph and find the center, the vertices, and the foci.

transverse axis: ◦ vertical

center:◦ (-2, 1)

vertices:◦ (-2, 3), (-2, -1)

foci:◦

2 2( 1) ( 2)1

4 9

y x

2 1 13( , )

Writing the Equation in Standard Form – You TryGiven 4x2 – 3y2 + 8x + 16 = 0You must complete the square

2 211

4 3

( )y x

EccentricityThe same formula applies to

both ellipses and hyperbolas. If the eccentricity is large, the

branches of the hyperbola are nearly flat.

If the eccentricity is close to 1, the branches are more narrow.

ce

a