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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.

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  • $\begingroup$ What angle do you have in mind? Isn't that the slope of the asymptote? $\endgroup$
    – Vasili
    Commented Dec 14, 2021 at 14:07
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    $\begingroup$ The axes of the hyperbola are the bisectors of the angles between the asymptotes. $\endgroup$ Commented Dec 14, 2021 at 14:08
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    $\begingroup$ Thank you. I've found it here: math.stackexchange.com/questions/2084774/… $\endgroup$
    – eMathHelp
    Commented Dec 14, 2021 at 14:15
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    $\begingroup$ 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. $\endgroup$ Commented Dec 14, 2021 at 16:37
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    $\begingroup$ In practice, the sign you must choose in that formula is the same as the sign of $${m_1p-q+b_1\over m_2p-q+b_2}.$$ $\endgroup$ Commented Dec 14, 2021 at 16:48

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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.

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