Geo4.102

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Find the equation passing through the origin and cutting the circles x^{2}+y^{2}-8y+12=0,x^{2}+y^{2}-4x-6y-9=0\, orthogonally.

Let the equation be

x^{2}+y^{2}+2gx+2fy+c=0\,

This circle passes through the origin,hence

c=0\,

The condition for the required circle and the first circle to cut orthogonally is

From the first circle, g_{1}=0,f_{1}=-4,c_{1}=12\,

2(g)(0)+2(f)(-4)=c+12\,

-8f=12,g={\frac  {-3}{2}}\,

The condition for the required circle and the second circle to cut orthogonally is

From the second circle, g_{2}=-2,f_{2}=-3,c_{2}=-9\,

2(g)(-2)+2({\frac  {-3}{2}}(-3)=c-9\,

-4g+9=-9,f={\frac  {9}{2}}\,

Hence the required circle is

x^{2}+y^{2}+9x-3y=0\,

Main Page:Geometry:Circles