A) \[2ax+2by-({{a}^{2}}+{{b}^{2}}+{{k}^{2}})=0\]
B) \[2ax+2by-({{a}^{2}}-{{b}^{2}}+{{k}^{2}})=0\]
C) \[{{x}^{2}}+{{y}^{2}}-3ax-4by+{{a}^{2}}+{{b}^{2}}-{{k}^{2}}=0\]
D) \[{{x}^{2}}+{{y}^{2}}-2ax-3by+({{a}^{2}}-{{b}^{2}}-{{k}^{2}})=0\]
Correct Answer: A
Solution :
Let the equation of circle x2 + y2 + 2gx + 2fy + c = 0, cuts the circle x2 + y2 =k2 orthogonally. \[\therefore \]\[2{{g}_{1}}{{g}_{2}}+2{{f}_{1}}{{f}_{2}}={{c}_{1}}+{{c}_{2}}\] \[\Rightarrow \]\[2g.0+2f.0=c-{{k}^{2}}\]\[\Rightarrow \]\[c={{k}^{2}}\] ?(i) Also,, x2 + y2 + 2ga+ 2fy + c = 0 passes through (a, b). \[\therefore \]\[{{a}^{2}}+{{b}^{2}}+2ga+2fb+c=0\] \[\therefore \]Required equation of locus of centre is, \[-2ax-2by+{{a}^{2}}+{{b}^{2}}+{{k}^{2}}=0\] \[\Rightarrow \]\[2ax+2by-({{a}^{2}}+{{b}^{2}}+{{k}^{2}})=0\]
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