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Coordinate geometry

  1. Dec 27, 2011 #1
    1. The problem statement, all variables and given/known data

    2 circles have the equation x2+y2-2x-2y+1=0 and x2+y2-12x-12y+36=0 respectively. Both circle touches the x-axis, y-axis and the line 3x + 4y = 12. Find the fourth tangent of the 2 circles.

    2. Relevant equations



    3. The attempt at a solution

    This is second part of the question, I solved the first part which require to find the equation of the 2 circles... And now i stuck at this final part... can anyone help me?
     
  2. jcsd
  3. Dec 27, 2011 #2

    ehild

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    The line 3x+4y is not tangent to the circles. Show the original problem, please.

    ehild
     
  4. Dec 27, 2011 #3
    Its 3x + 4y = 12.... not 3x+4y=0...
     
  5. Dec 27, 2011 #4

    ehild

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    Well, I wanted to say 3x+4y=12. Check if it is tangent to the circles.

    ehild
     
  6. Dec 27, 2011 #5

    micromass

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    I find that it is tangent to the circles. Perhaps you miscalculated??

    But it's a good starter question to the OP: how do you check whether a line is tangent to the circle??
     
  7. Dec 28, 2011 #6
    If a line is tangent to the circle, then the perpendicular distance from the center of the circle to the given line is equals to the radius of the circle...

    So what is the trick to find fourth tangent of the 2 circles? I cannot figure out how to find it..:confused:
     
  8. Dec 28, 2011 #7

    ehild

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    The common tangent line has one common point with both circles. So the equation of a circle and that of the line have a single solution.
    Draw those circles. The symmetry of the figure gives you hint about the other tangent.

    ehild
     
    Last edited: Dec 28, 2011
  9. Dec 28, 2011 #8

    ehild

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    Thank you, micromass!
    You are right, my calculator played tricks with me. It IS a tangent line.

    ehild
     
  10. Dec 28, 2011 #9

    SammyS

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    Graphing the two circles may lead one to think that the circles share a point of tangency. I did just that using Wolfram Alpha. I then included the given line, 3x + 4y = 12, and zoomed-in.

    attachment.php?attachmentid=42254&stc=1&d=1325083626.gif

    attachment.php?attachmentid=42256&stc=1&d=1325083606.gif
     

    Attached Files:

  11. Dec 28, 2011 #10

    ehild

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    Nice pictures SammyS!

    A pair of tangent lines to two circles are mirror images of each other to the line that connects the centre of the circles.
    The centres of the circles (x-1)2+(y-1)2=1 and (x-6)2+(y-6)2=36 are (1,1) and (6,6); both lie on the y=x line. So a graph of the circles and the tangent lines is invariant when x and y are exchanged. With the change x<=>y in the equation of a tangent line, you get the other one.

    ehild
     
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