How do you write the equation of a hyperbola in standard form given Foci: (3,+-2) and Asymptotes: y = +-2(x-3)?

1 Answer
Narad T.
Oct 31, 2016

The equation of the hyperbola is #(y)^2/(16/5)-(x-3)^2/(4/5)=1#

Explanation:

The standard equation of a hyperbola is
#(y-k)^2/b^2-(x-h)^2/a^2=1#
The slopes of the asymptotes are #+-b/a#
so #+-b/a=+-2# #=>##b=2a#
The foci are #h,k+-c#
So #h=3#
and #k+c=2# and #k-c=-2#
So from the eqautions #k=0# and #c=2#
#c^2=a^2+b^2#
Therefore #4=a^2+4a^2=5a^2# #=># #a^2=4/5#
And #b^2=4a^2=16/5#

So the equation of the hyperbola is
#(y)^2/(16/5)-(x-3)^2/(4/5)=1#

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