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Math Help - Circles

  1. #1
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    Circles

    The circle S_1 with center C_1 (a_1,b_1) and radius r_1 touches externally the circle s_2 center C_2 (a_2,b_2) radius r_2. The tangent at their common point passes through the origin. Show that,
    ({a_1}^2-{a_2}^2)+({b_1}^2-{b_2}^2)=({r_1}^2-{r_2}^2)
    If also, the other two tangents from the origin to S_1 and S_2 are perpendicular, prove that |a_2b_1-a_1b_2|=|a_1a_2+b_1b_2|.
    Hence show that if C_1 remains fixed but S_1 and S_2 vary, then C_2 lies on the curve
    ({a_1}^2-{b_1}^2)(x^2-y^2)+4a_1b_1xy=0.
    I'm quite sure if I know how to do the first part, I should be able to do the rest, but I don't even know how to start. Thanks for any help.
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  2. #2
    Senior Member pacman's Avatar
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    Arze: "The circle with center and radius touches externally the circle center radius . The tangent at their common point passes through the origin. Show that,
    "

    Solution for (i) if they have a common point of (0,0), use the standard formula for circle:

    (0 - A)^2 + (0 - B)^2 = R^2,
    (0 - a)^2 + (0 - b)^2 = r^2,

    subtract,

    (A^2 - a^2) + (B^2 - b^2) = (R^2 - r^2)
    Attached Thumbnails Attached Thumbnails Circles-circg.gif  
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  3. #3
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    but how do we know that the point is (0,0)?
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