A) An ellipse
B) The radical axis of the given circles
C) A conic
D) Another circle
Correct Answer: B
Solution :
Let the circle be\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]. This cuts the two given circles orthogonally, therefore \[2(g{{g}_{1}}+f{{f}_{1}})=c+{{c}_{1}}\] ?.(i) and \[2(g{{g}_{2}}+f{{f}_{2}})=c+{{c}_{2}}\] ?.(ii) Subtracting (ii) from (i), we get \[2g({{g}_{1}}-{{g}_{2}})+2f({{f}_{1}}-{{f}_{2}})={{c}_{1}}-{{c}_{2}}\] So locus of \[(-g,\ -f)\]is \[-2x({{g}_{1}}-{{g}_{2}})-2y({{f}_{1}}-{{f}_{2}})={{c}_{1}}-{{c}_{2}}\] or \[2x({{g}_{1}}-{{g}_{2}})+2y({{f}_{1}}-{{f}_{2}})+{{c}_{1}}-{{c}_{2}}=0\], which is the radical axis of the given circles.You need to login to perform this action.
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