Geo4.134

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Find the coaxal system which is orthogonal to the coaxal system x^{2}+y^{2}+2x-3y-1+k(x-4y+1)=0\,

Given equation of the coaxal system is

x^{2}+y^{2}+2x-3y-1+k(x-4y+1)=0\,

x^{2}+y^{2}+x(2+k)-y(3+4k)+k-1=0\,

Let the equation to the circle orthogonal to the above circle is

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

Now

g(2+k)+f(-3-4k)=c+k-1\,

k(g-4f-1)+2g-3f+1-c=0\, for all k.

hence

g-4f-1=0,2g-3f+1-c=0\,

8f+2-3f+1=c,5f=c-3,f={\frac  {c-3}{5}}\,

g=4({\frac  {c-3}{5}})+1\,

Substituting these values in the required equation,

x^{2}+y^{2}+8{\frac  {c-3}{5}}+2+2{\frac  {c-3}{5}}+c=0\,

Simplifying, we get

5x^{2}+5y^{2}-14x-6y+c(8x+2y+5)=0\,


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