Geo4.94

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Find the equation of the circle which passes through the points of intersection of x^{2}+y^{2}=4,x^{2}+y^{2}-2x-4y+4=0\, and touch the line x+2y=5\,

Equation of the common chord is

(x^{2}+y^{2}-4)-(x^{2}+y^{2}-2x-4y+4)=0\,

2x+4y-8=0\,

x+2y-4=0\,

Equation of the circle passing through the point of intersection of the two circles is

x^{2}+y^{2}-4+k(x+2y-4)=0\,

x^{2}+y^{2}+kx+2ky+(-4-4k)=0\,

Centre of the circle is

({\frac  {-k}{2}},-k)\,

Radius is

{\frac  {{\sqrt  {k^{2}+4k^{2}+16+16k}}}{2}}\,

Given that the circle touches the line x+2y-5=0\,

{\frac  {|-k-4k-10|}{{\sqrt  {5}}}}={\frac  {{\sqrt  {k^{2}+4k^{2}+16+16k}}}\,}

{\sqrt  {5}}(k+2)={\sqrt  {5k^{2}+16k+16}}\,

5k^{2}+20+20k=5k^{2}+16k+16\,

4k+4=0\,

k=-1\,

Required Equation of the circle is

x^{2}+y^{2}-4-(x+2y-4)=0\,

x^{2}+y^{2}-x-2y=0\,

Main Page:Geometry:Circles