Find angle of rotation of hyperbola given two asymptotes

Question

Given two asymptotes $y=m_1x+b_1$ and $y=m_2x+b_2$, and a point on hyperbola $(p,q)$ is there a formula for finging an angle of rotation?

I've found a formula (How do I find the slope of an angle bisector, given the equations of the two lines that form the angle?) for finding two possible lines that contain the major axis, but I don't how to use the point to find unique line.

What angle do you have in mind? Isn't that the slope of the asymptote? — Vasili, Dec 14 '21 at 14:07
The axes of the hyperbola are the bisectors of the angles between the asymptotes. — Intelligenti pauca, Dec 14 '21 at 14:08
Thank you. I've found it here: https://math.stackexchange.com/questions/2084774/how-do-i-find-the-slope-of-an-angle-bisector-given-the-equations-of-the-two-lin — eMathHelp, Dec 14 '21 at 14:15
Point $(p,q)$ lies in one of the four angles formed by the asymptotes: choose as major axis the bisector lying in the same angle. — Intelligenti pauca, Dec 14 '21 at 16:37
In practice, the sign you must choose in that formula is the same as the sign of $${m_1p-q+b_1\over m_2p-q+b_2}.$$ — Intelligenti pauca, Dec 14 '21 at 16:48

score 1 · Answer 1 · answered Dec 14 '21 at 16:47

1

I got it. I just need to take a point on a bisector and check whether it has lies in the same (or opposite) quadrant as the given point.

answered Dec 14 '21 at 16:47

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1 Answers1