# Geo4.135

Show that as k varies the circles$x^{2}+y^{2}+2ax+2by+2k(ax-by)=0\,$ form coaxal system.Find the radical axis.

Given circles are $x^{2}+y^{2}+2ax+2by+2k(ax-by)=0\,$. Equation 1

Two members of the family are $x^{2}+y^{2}+2ax+2by+2k_{1}(ax-by)=0,x^{2}+y^{2}+2ax+2by+2k_{2}(ax-by)=0\,$

Let these be 2 and 3

(2)-(3) gives

$2(k_{1}-k_{2})(ax-by)=0,ax-by=0\,$

Equation 1 represents a system of coaxal circles whose radical axis is $ax-by=0\,$

It can be written as

$x^{2}+y^{2}+2a(1+k)x+2b(1-k)y=0\,$

Let $x^{2}+y^{2}+2gx+2fy+c=0\,$ cut this circle orthogonally,

Therefore

$2g(1+k)a+2fb(1-k)=c+0\,$

Simplifying this equation, we have

$2(ag-bf)k+2(ag+bf)=c\,$

If this is true for all values of k, then we have $ag-bf=0,ag+bf={\frac {c}{2}}\,$

Solving

$g={\frac {c}{4a}},f={\frac {c}{4b}}\,$

Therefore, the system of circles orthogonal to (1) is $x^{2}+y^{2}+{\frac {c}{2a}}x+{\frac {c}{2b}}y+c=0\,$

$2ab(x^{2}+y^{2})+c(bx+ay+2ab)=0\,$

$ab(x^{2}+y^{2})+k(bx+ay+2ab)=0\,$