Geo4.123

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Find the limiting points of the coaxal sustem determined by the circles x^{2}+y^{2}+5x+y+4=0,x^{2}+y^{2}+10x-4y-1=0\,

Equation to the radical axis (x^{2}+y^{2}+5x+y+4)-(x^{2}+y^{2}+10x-4y-1)=0\,

-5x+5y+5=0\,

x-y-1=0\,

Any circle of coaxal system is

x^{2}+y^{2}+5x+y+4+k(x-y-1)=0\,

x^{2}+y^{2}+x(5+k)+y(1-k)+4-k=0\,

If this is a limiting point circle,radius is zero.

{\sqrt  {[{\frac  {10+k}{2}}]^{2}+[{\frac  {4+k}{2}}]^{2}+1+k}}=0\,

k^{2}+16k+60=0\,

(k+6)(k+10)=0\,

k=-6,-10\,

Hence the limiting points of the circle are given by the centre \left(-{\frac  {10+k}{2}},{\frac  {4+k}{2}}\right)\,

Therefore the limiting points are

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


Main Page:Geometry:Circles