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Find the condition that the two circles x^{2}+y^{2}+2a_{1}x+2b_{1}y=0\, and x^{2}+y^{2}+2a_{2}x+2b_{2}y=0\, touch eachother.

Let the equations of the two circles are


There is no constant term in both the equations. So both the circles touch each other at O(0,0).

Centre of the circles are A(-a_{1},-b_{1}),B(-a_{2},-b_{2})\,

The circles touch eachother. A,O,B are collinear

Hence Slope of AO=Slope of BO

{\frac  {b_{1}}{a_{1}}}={\frac  {b_{2}}{a_{2}}}\,


This is the required condition.

Main Page:Geometry:Circles