Geo4.135

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Show that as k varies the circlesx^{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\,


Main Page:Geometry:Circles