# Finding the equation of a hyperbola that has two arbitrary intersecting lines as asymptotes

I have searched through math.stackexchange for related posts, but failed to either connect or transfer/map to my following mind-experiment:

Suppose we are given two lines described by: $$3x+4y=2$$ and further $$-3x-2y=2$$. They both intersect at $$(-2,2)$$. How can I compute the hyperbola using these two lines as the hyperbolas asymptotes?

A simple first naive approach with $$\frac{(x-x_0)^2}{a^2}-\frac{(y-y_0)^2}{b^2}=1$$ did not yield the expected result. It seems like the rotation is missing.

Thank you in advance for any hints and With best regards

• there could be two hyperbolas with those asymptotes – J. W. Tanner Nov 3 '19 at 18:29
• Actually, there are infinitely-many hyperbolas with a given pair of asymptotes. Consider: $xy=k$, for any $k$, is a hyperbola with the coordinate axes as asymptotes. – Blue Nov 3 '19 at 18:35
• It's worth noting that the product of the distances from a point on a hyperbola to the asymptotes is a constant. So, if you know the formula for the distance from a point to a line ... – Blue Nov 3 '19 at 18:46

The equation of a hyperbola whose asymptotes have equations $$3x+4y-2=0$$ and $$-3x-2y-2=0$$ can be written as: $$(3x+4y-2)(-3x-2y-2)=k,$$ where $$k$$ is a constant. For every value of $$k$$ you'll get a different hyperbola. Given any point $$P$$ in the plane, you can choose $$k$$ such that the hyperbola passes through $$P$$.