# coordinate geometry

• Jul 12th 2010, 07:18 AM
prasum
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
• Jul 12th 2010, 01:22 PM
Opalg
Quote:

Originally Posted by prasum
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 $g_1g_2>0$.