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Math Help - Find the best way to solve this question

  1. #1
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    Find the best way to solve this question

    Find the equation of the circle which is tangent to the line 3x - y - 1 = 0 ~ at  A(1,2)~ and passes through the point  B(5,-2)

    I've got the best way to solve this problem , do you know how i solved it ?
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Well, I did the usual textbook stuff

    Spoiler:
    First I found the equation of the line normal to 3x-y-1=0 and passing through A. The centre of the circle lies on this line. Then I found the equation of the perpendicular bisector of the chord AB. The centre of the circle also lies on this line. Solving for the point of intersection, I determined the centre of the circle to be (4,1). Finally, I found the radius of the circle, which is \sqrt{10}. Hence the circle is (x-4)^2+(y-1)^2=10.

    Nevertheless the method did not appear to be particularly quick, so I guess you have a quicker solution.
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  3. #3
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    Let F be a family of circles such that all members of the family F touch the line  3x - y - 1 = 0 ~~at  A(1,2) .

    Then F should be  (x-1)^2 + (y-2)^2 + k(3x - y - 1) = 0 .

    Since the required member passes through  B(5,-2) ,

    it satisifies  (5-1)^2 + (-2-2)^2 + k[3(5) - (-2) - 1] = 0 .

    Therefore ,  k = -2

    Finally substitute back to F ,

    The required equation is :

     (x-1)^2 + (y-2)^2 -2(3x-y-1) = 0
     x^2 + y^2 - 8x - 2y + 7 = 0
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