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Show smaller circle lie inside bigger circle

  1. Jul 14, 2015 #1
    1. The problem statement, all variables and given/known data

    I was asked to show the circle $x^2$ + $y^2$ +6$x$ -10$y$ +9=0 lies entirely inside circle $x^2$ + $y^2$ +4$x$ -6$y$-48=0...
    For circle $x^2$ + $y^2$ +6$x$ -10$y$ +9=0
    I managed to get the centre = (3,5) r=5

    For circle $x^2$ + $y^2$ +4$x$ -6$y$-48=0
    i gt centre = (2,3) r= sqrt 61


    How to proceed from here ??

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Jul 14, 2015 #2

    phinds

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    Draw them on a piece of graph paper.
     
  4. Jul 14, 2015 #3

    RUber

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    If you are looking to show it analytically, you could look for points of intersection by solving both equations for one variable and setting them equal. That seems like too much work for this problem though.
    Also you could use the radius of the larger circle. Find the radius that passes through the center of the smaller circle. If the sum of the distance between the centers and the radius of the smaller circle is less than the radius of the larger, then you know it will be entirely inside the larger circle.
     
  5. Jul 14, 2015 #4

    haruspex

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    Are you sure? I think it might be the easiest way, but not by solving either in isolation. You can get a linear equation very quickly.
     
  6. Jul 15, 2015 #5

    BvU

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    It doesn't matter for this exercise, but your center points are (-3,5) and (-2,3)...

    Also, if you draw two circles, one inside the other (and preferably not concentric :smile: ), you quickly see that the points of closest approach are on a line through the centers. And you can comfortably see what RUber means. (Comfortably meaning: it's easier to see than to describe in words)
     
  7. Jul 16, 2015 #6

    ehild

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    See picture. If you shrink both circles by the radius of the smaller one, the small circle becomes a point and you need to figure out if it is inside the shrunk big circle.
    circleincircle.JPG
     
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