Geo4.122

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The line x+2y-1=0\, is the radical axis and the circle x^{2}+y^{2}=1\, is a member of a coaxal system.Find the circle touching the line x+2y=0\, and belonging to the system.

The circle of given coaxal system is

x^{2}+y^{2}+k(x+2y-1)=0\,

x^{2}+y^{2}-1+kx+2ky-k=0\,

Centre is ({\frac  {-k}{2}},-k)\, and radius is{\sqrt  {{\frac  {k^{2}+4k^{2}+4k+4}{4}}}}\,

This circle touches the line x+2y=0\,

Hence

{\frac  {|-k-4k|}{2{\sqrt  {5}}}}={\sqrt  {{\frac  {5k^{2}+4k+4}{4}}}}\,

25k^{2}=25k^{2}+20k+20\,

k=-1\,

The required circle is x^{2}+y^{2}-x-2y=0\,

Main Page:Geometry:Circles