Geo4.74

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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

S_{1}=x^{2}+y^{2}+2a_{1}x+2b_{1}y=0,S_{2}=x^{2}+y^{2}+2a_{2}x+2b_{2}y=0\,

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}}}\,

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

This is the required condition.

Main Page:Geometry:Circles