SOLUTION: A hyperbola with a horizontal transverse axis passes through the point (0,3). The equations of the asymptotes are y=x+1 and y=5-x. Determine the equation of the hyperbola.

Question 1185475: A hyperbola with a horizontal transverse axis passes through the point (0,3). The equations of the asymptotes are y=x+1 and y=5-x. Determine the equation of the hyperbola.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

With a horizontal transverse axis, the general equation is

(h,k) is the center of the hyperbola
2a is the length of the transverse axis (i.e., a is the distance from the center to each vertex)
2b is the length of the conjugate axis
The slopes of the asymptotes are b/a and -b/a

To find the center (h,k), note that the two asymptotes intersect at the center of the hyperbola. Solve the given pair of equations of the asymptotes to find h and k.

You should find k=3, which means the given point (0,3) is one of the vertices of the hyperbola. So the distance from the center (h,k) to the vertex (0,3) is a -- half of the transverse axis.

Now you know a, so you know a^2; knowing the slopes 1 and -1 are b/a and -b/a, you can determine b^2.

And now you have all the pieces you need to write the equation.

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ANSWER (but if you want to learn anything from this you should go through the steps described above to find it):

A graph, showing the two branches of the hyperbola and the two asymptotes....

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