Geo4.124

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Find the limiting points of the coaxal sustem determined by the circles x^{2}+y^{2}+2x+4y+7=0,x^{2}+y^{2}+4x+2y+5=0\,

The equation of the radical axis is

(x^{2}+y^{2}+2x+4y+7)-(x^{2}+y^{2}+4x+2y+5)=0\,

-2x+2y+2=0\,

x-y-1=0\,

Equation of a circle of coaxal system is

x^{2}+y^{2}+2x+4y+7+k(x-y-1)=0\,

x^{2}+y^{2}+2x(1+{\frac  {k}{2}})+2y(2-{\frac  {k}{2}})+(7-k)=0\,

If this is a limiting point circle,radius is zero

{\sqrt  {[-(1+{\frac  {k}{2}})]^{2}+[-(2-{\frac  {k}{2}})]^{2}-7+k}}=0\,

(4+k^{2}+4k+16+k^{2}-8k-28+4k)=0\,

2k^{2}-8=0\,

k=\pm 2\,

Hence the limiting points are

(-2,-1),(0,-3)\,

Main Page:Geometry:Circles