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B (solved) Given circle, find the line of bearing of tan lines thru 0,0

  1. Nov 4, 2016 #1
    This question occurred to me a few days ago and it's been bugging me ever since.

    Consider a circle in the coordinate plane with center P(x,y) and radius R, where R < D, D being the distance from the origin to the circle's center.

    There are two lines tangent to the circle (T1 and T2) that pass through the origin.

    The circle's location can also be considered as a line segment of radius D and some angle Θ with the x axis.

    How can I find a general equation in terms of x, y, R, and Θ to determine the angle each tangent line makes with the x axis?

    073G7O0.png

    Solution found:

    C9qMSEA.png

    Adding Θs to Θdx should yield the bearing for the other tangent line.
     
    Last edited: Nov 4, 2016
  2. jcsd
  3. Nov 4, 2016 #2

    jedishrfu

    Staff: Mentor

    You might consider that where the line hits the circle it will be perpendicular to some radius so you have some D length and some angle that D makes with the X axis and some trig to mix in to get your answer.
     
  4. Nov 4, 2016 #3
    Solved. The help is appreciated.
     
    Last edited: Nov 4, 2016
  5. Nov 4, 2016 #4
    I can't tell if this is just a coincidence of the values I chose, but the distance from the x-axis to the circle's center (Dy) worked out to be equal to the distance from the origin to the tangent line's intersection (T). Weird.
     
  6. Nov 4, 2016 #5

    jedishrfu

    Staff: Mentor

    This may be a theorem and might be proven if you think about it more abstractly. You are actually saying that drawing a circle with center at the origin that passes through the original circle's center will intersect at the tangent line points.

    Perhaps @Mark44 could verify your observation and point to the theorem if it exists as I don't know for sure.

    Edit: okay I think this is it on Wikipedia

    https://en.m.wikipedia.org/wiki/Tangent_lines_to_circles

    Look down a bit in the article to see two intersecting circles and how tangent lines from one circle go through the center of another.
     
  7. Nov 4, 2016 #6

    Thanks, this is interesting. Also I just noticed an error in my second diagram when I renamed my variables: Θty should beΘtx everywhere except in the cutout triangle, where it's correctly labeled. I'm pretty sure the calculations themselves are still correct. I'll update the image.

    Never mind, the OP no longer has an edit option. I'll just post it here.

    WIw8iv5.png
     
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