Geo4.129

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Find the equation of the circle which belongs to the coaxal system determined by (0,-3) and (-2,-1) and which is orthogonal to the circle x^{2}+y^{2}+2x+6y+1=0\,

The equations of the limiting point circles are

(x-0)^{2}+(y+3)^{2}=0,(x+2)^{2}+(y+1)^{2}=0\,

x^{2}+y^{2}+6y+9=0,x^{2}+y^{2}+4x+2y+5=0\,

Equation of the radical axis is

4x-4y-4=0,x-y-1=0\,

Equation of the coaxal circle is

x^{2}+y^{2}+6y+9+k(x-y-1)=0\,

x^{2}+y^{2}+2kx+2y(3-{\frac  {k}{2}})+9-k=0\,

This circle cuts the given circle orthogonally, hence

2(1)({\frac  {k}{2}})+2(3)({\frac  {6-k}{2}})=9-k+1\,

By simplifying,we get

k=8\,

Therefore,the equation to the required circle is

x^{2}+y^{2}+8x-2y+1=0\,


Main Page:Geometry:Circles