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Need Help With Geometry Problem (Circles)

  1. Apr 18, 2007 #1
    In the diagram below:
    AB is the diameter of the semicircle with center O. Circles P and Q are tangent to each other and to the semicircle. If OB=4, find the radius of circle Q.

    I haven't been able to make any headway at all with this problem. I tried to find a system of equations with the radius of circle Q equal to x and some other length equal to y, but all I found was that the length of the common external tangent of circles P and Q is [tex] 2 \sqrt{2x} [/tex], where x is the radius of circle Q, and I'm not sure how that's useful. Please help.
    Last edited: Apr 18, 2007
  2. jcsd
  3. Apr 18, 2007 #2


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    Let the origin be (0,0) in a cartesian coordinate system. The point P is then (0,2). Let the point Q be at position (x,y).

    To define the position of a circle needs three equations. (There are 3 degrees of freedom, the position (x,y) of the center, and the radius.)

    The radius of Q is y, this is one restriction.

    The circle Q and P are tangent. This means that the distance from P to Q is equal to the sum of their radii.

    The third restriction on the circle centered at Q is that it be tangent to the circle centered at O. This will be a quadratic equation in x and y.
  4. Apr 18, 2007 #3
    I'd already figured out the first two restrictions, and I understood that the third restriction would have to do with circle Q being tangent to the semicircle, but I have no idea how to derive an equation from that.

    EDIT: Never mind. I figured it out. Thank you.
    Last edited: Apr 18, 2007
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