9.5. Hyperbolas http://www.ck12.org
Example B
Find the equation of the hyperbola with foci at (-3, 5) and (9, 5) and asymptotes with slopes of±^43.
Solution:The center is between the foci at (3, 5). The focal radius isc=6. The slope of the asymptotes is always
the rise over run inside the box. In this case since the hyperbola is horizontal andb ais in thexdirection the slope is
a. This makes a system of equations.
b
a=±4
3
a^2 +b^2 = 62When you solve, you geta=
√
13 ,b=^43√
13.
(x− 133 )^2 −(y 16 − 5 )^2
9 ·^13 =^1
Example C
Find the equation of the conic that has a focus point at (1, 2), a directrix atx=5, and an eccentricity equal to^32. Use
the property that the distance from a point on the hyperbola to the focus is equal to the eccentricity times the distance
from that same point to the directrix:
PF=ePD
Solution:This relationship bridges the gap between ellipses which have eccentricity less than one and hyperbolas
which have eccentricity greater than one. When eccentricity is equal to one, the shape is a parabola.√
(x− 1 )^2 +(y− 2 )^2 =^32√
(x− 5 )^2
Square both sides and rearrange terms so that it is becomes a hyperbola in graphing form.