Geo4.133

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Find the equation to the system of circles orthogonal to the coaxal system x^{2}+y^{2}+3x+4y-2+k(x+y-7)=0\,

Equation to the given coaxal system is

x^{2}+y^{2}+3x+4y-2+k(x+y-7)=0\,

x^{2}+y^{2}+x(3+k)+y(4+k)-2-7k=0\,

Let the equation of the circle orthogonal to the given system be

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

Hence

g(3+k)+f(a4+k)=c-2-7k\,

k(g+f+7)+3g+4f+2-c=0\, for all k.

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

Solving these two,

g=-26-c,f=19+c\,

Now the equation becomes

x^{2}+y^{2}-2x(26+c)+2y(19+c)+c=0\,

x^{2}+y^{2}-52x+38y-c(2x+2y+1)=0\, which is the required orthogonal system, where c is a parameter.


Main Page:Geometry:Circles