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Math Help - Calculate Tangent Line to 2 Circles

  1. #1
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    Calculate Tangent Line to 2 Circles

    I am trying to calculate the equation of line tangent to 2 circles. I have attached a picture to give a better idea.

    Circle 1 has a center at (3,4) and radius of 5
    Circle 2 has a center at (9,19) and radius of 11,

    So far I have wrote the equation = of the 2 circles, and found the distance between the points the tangent lines touch on each circle.

    My picture is not the most acurate one, it is just to give an idea.

    Thanks for the help
    Attached Thumbnails Attached Thumbnails Calculate Tangent Line to 2 Circles-problem.jpg  
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  2. #2
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    Geometry of tangents

    First, remember that a tangent to a circle is perpendicular to a radial line segment to the point of tangency. Thus, recalling how we define the distance between a point and a line, the distance between a line tangent to a circle and the center of that circle is the radius of that circle. For a line of the form Ax+By+C=0, the distance to the point (x_0,y_0) is d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}
    Since the centers of the circles are (3,4) and (9,19), and the radii are 5 and 11, respectively, you have
    \frac{|3A+4B+C|}{\sqrt{A^2+B^2}}=5
    and
    \frac{|9A+19B+C|}{\sqrt{A^2+B^2}}=11

    Note you will have an unneeded degree of freedom (as you could multiply the line equation Ax+By+C=0 by a constant and still have the same line), so you can pick A,B,C to be scaled such that A^2+B^2=1, so you then can define \theta such that A=\cos\theta,\,\,B=\sin\theta, and our equations are
    |3\cos\theta+4\sin\theta+C|=5
    and
    |9\cos\theta+19\sin\theta+C|=11

    Thus you have two equations for two unknowns. Note that there will be four solutions: two will be the 'external' tangents, and the other two will be 'internal' tangents (which cross each other between the circles)

    --Kevin C.
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  3. #3
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    Quote Originally Posted by CaliMan982 View Post
    I am trying to calculate the equation of line tangent to 2 circles. I have attached a picture to give a better idea.

    Circle 1 has a center at (3,4) and radius of 5
    Circle 2 has a center at (9,19) and radius of 11,

    So far I have wrote the equation = of the 2 circles, and found the distance between the points the tangent lines touch on each circle.

    My picture is not the most acurate one, it is just to give an idea.

    Thanks for the help
    Hello,

    I've attached a slightly more accurate sketch.

    The steps to do the construction:

    1. Draw a circle around the center of the green circle with radius r_{green}-r_{blue}
    2. Draw a circle around the midpoint of M_{green}M_{blue}
    3. Connect the intersection points of the circle of #2 with the circle of #1 with M_{blue}. The tangent you are looking for is a parallel of this line.
    4. Translate this line by r_{blue} units until it touches both circles.

    Calculating the angles:

    \tan(\alpha) = \frac{15}6 = \frac52

    For symmetry reasons the angles \beta_1 and \beta_2 must be equal:

    \tan(\beta) = \frac6{15} = \frac25

    Calculating the slop of the tangents (red):

    Use \tan(\alpha + \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha) \cdot \tan(\beta)}

    Now you have to calculate the coordinates of the touching points on one of the circles to complete the equation of the tangent.
    Attached Thumbnails Attached Thumbnails Calculate Tangent Line to 2 Circles-zweikrs_zweitang.gif  
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  4. #4
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    where does the 16/5 com from?
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  5. #5
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    Quote Originally Posted by CaliMan982 View Post
    where does the 16/5 com from?
    Pardon?

    If you mean \frac{15}{6} then it is the slope between the 2 centres of the circles:

    \frac{19-4}{9-3}=\frac{15}{6}
    Last edited by earboth; January 23rd 2008 at 09:49 PM. Reason: typo
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  6. #6
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    How would I calculate a point on the cirlce the tangent line touches, i just calculated the slope.
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  7. #7
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    Quote Originally Posted by CaliMan982 View Post
    How would I calculate a point on the cirlce the tangent line touches, i just calculated the slope.
    Have a look here: http://www.mathhelpforum.com/math-he...348-post1.html
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  8. #8
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    Complete solution to the four tangent lines of two circles

    Tangents to Two Circles gives complete expressions for cos theta, sin theta, and "C" in terms of the coordinates of the centers and radii of the two circles, which you may find helpful.
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