Geo4.108

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Find the equation of the circle which cuts orthogonally three circles x^{2}+y^{2}-x+y=0,2x^{2}+2y^{2}+5x-3y-2=0\, and x^{2}+y^{2}+4x-2y-2=0\,

Let the equation of the required circle be

x^{2}+y^{2}+2gx+2fy+c=0\,.Let this equation be 1.

The given circles be 2,3,4.

Equation 1 cuts 2,3,4 orthogonally,

2g({\frac  {-1}{2}})+2f({\frac  {1}{2}})=c+0,-g+f=c,f=g+c\,. equation 5.

2g({\frac  {5}{4}})+2f({\frac  {-3}{4}})=c-1,5g-3f=2c-2\,. equation 6

2g(2)+2f(-1)=c-2,4g-2f=c-2\,. equation 7.

Substituting 5 in 6 and 7

5g-3(g+c)-2c+2=0,4g-2(g+c)-c+2=0\,

2g-5c+2=0,2g-3c+2=0\,

-5c+3c=0,c=0\,

2g+0=-2,g=-1,f=-1\,

The equation of the required circle is

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

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


Main Page:Geometry:Circles