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Math Help - Two tangent lines to a circle

  1. #1
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    Two tangent lines to a circle

    Given two intersecting lines of the form y = mx+c, which are both tangential to a circle of radius r, how do I find an algebraic solution for:

    • the location of the circle centre
    • the location of each line/circle intersection

    I guess there are two possible answers, as the circle could be in one of two positions.

    Many thanks,
    Siskin
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint :

    If r_1 and r_2 are the given lines, the center C of any circle tangential to r_1 and r_2 satisfies

    d ( C , r_1 ) = d ( C , r_2 ) = r .
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  3. #3
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    Give the line equations and radius.Without definition there are infinite solutions.


    bjh
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  4. #4
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    Quote Originally Posted by siskin View Post
    Given two intersecting lines of the form y = mx+c, which are both tangential to a circle of radius r, how do I find an algebraic solution <--- we need the equations of the lines and the length of the radius to help you for:

    • the location of the circle centre
    • the location of each line/circle intersection

    I guess there are two possible answers, as the circle could be in one of two <--- actually there are 4 possible positions positions.

    Many thanks,
    Siskin
    Let \alpha denote the acute angle between the 2 lines and e the distance between the point of intersection of the 2 lines and the center of a circle.

    Then you know that

    sin( \alpha/2)= r/e

    and

    cos( \alpha/2) = r/e

    (see attachment)
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  5. #5
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    I view this problem as a geometry problem with trig not allowed.A coordinate geometry solution could be described but it would seem that a geometry student would be given the required data and asked for answers.


    bjh
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