How do you write an equation for the hyperbola with center (0,0), vertex (-2,0) and focus (4,0)?

1 Answer
Douglas K.
Oct 30, 2016

Please see the explanation.

Explanation:

The given, center, vertex, and focus share the same y coordinate, 0, ,therefore, the standard form for the equation of this type of hyperbola is the one corresponding to the Horizontal Transverse Axis type:

#(x - h)^2/a^2 - (y - k)^2/b^2 = 1#

where #(h, k)# is the center, #a# is distance from the center to the vertex, and #b# affects the distance, #c#, from the center to the focus as determined by the equation #c^2 = a^2 + b^2#.

Substitute the center #(0, 0)# into the standard form:

#(x - 0)^2/a^2 - (y - 0)^2/b^2 = 1#

We know that #a = 2#, because one of the vertices a distance of 2 from the center. Substitute 2 for a into the equation:

#(x - 0)^2/2^2 - (y - 0)^2/b^2 = 1#

Observing that the focus is a distance of 4 from the center we set #c^2 = 16# in the equation #c^2 = a^2 + b^2#:

#16 = a^2 + b^2#

Substitute 4 for #a^2#

#16 = 4 + b^2#

Solve for b:

#b= sqrt(12)#

Substitute #sqrt(12)# for b into the equation:

#(x - 0)^2/2^2 - (y - 0)^2/(sqrt(12))^2 = 1#

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