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Math Help - Circle problem

  1. #1
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    Circle problem

    Can anyone help me solve this question.I cannot seem to figure it out.

    Q.Find an equation of the chord of contact of the tangents drawn from (4,5) to the circle:
    2x^{2}+2y^{2}-8x+12y+21=0

    Thanks
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  2. #2
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    Re: Circle problem

    Hi,
    You just have to remember how to construct the tangent lines to a circle from a point outside the circle. Then just apply some simple analytic geometry and algebra. The attachment shows most of the details:

    Circle problem-mhfgeometry60.png
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  3. #3
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    Re: Circle problem

    Hey Hacker 804
    the question you asked refers to the well known problem of the polar line of point P(4,5) to the conic ( in your case the given circle)
    See here: Pole and polar - Wikipedia, the free encyclopedia
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  4. #4
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    Re: Circle problem

    I forgot to give you the general form of the quation of the polar line.

    If P(a,b) is any point outside of the circle x^2+y^2+Ax+By+C=0 then the equation of the polar line is ax+by+A(x+a)/2+B(y+b)/2 +C=0 .
    In your case a=4,b=5 A=-4 B= 6 and C=21/2 .Apply this to find directly he equation that John stated in his post....
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  5. #5
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    Re: Circle problem

    Hi again,
    My previous response was an "elementary" solution. As Minoanman points out, if you know about inversion with respect to a circle, you can give a general formula for the desired line. The attachment gives such a formula (different from but equivalent to Minoanman's formula):

    Circle problem-mhfgeometry60b.png
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