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Math Help - coordinate geometry

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    coordinate geometry

    if one of the circles x^2+y^2+2g1x+c=0 and x^2+y^2+2g2x+c=0 lies within the other then prove that g1g2>0 and c>0
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    Quote Originally Posted by prasum View Post
    if one of the circles x^2+y^2+2g_1x+c=0 and x^2+y^2+2g_2x+c=0 lies within the other then prove that g_1g_2>0 and c>0
    If c\leqslant0 then the point (0,\sqrt{-c}) lies on both circles. In that case, the circles intersect, so one of them cannot lie inside the other. Conclusion: c must be greater than 0.

    So now assume that c>0. Check that in this case, neither circle contains any points on the y-axis. So each circle (and its interior) lies entirely on one side of the y-axis. Your next step is to find where the centres of the circles are. If they are on opposite sides of the y-axis then neither circle can lie inside the other. So the centres must lie on the same side of the y-axis. Deduce that g_1 and g_2 have the same sign and therefore g_1g_2>0.
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