A) \[gx+fy+c({{x}^{2}}+{{y}^{2}})\]
B) \[{{(gx+fy)}^{2}}={{x}^{2}}+{{y}^{2}}\]
C) \[{{(gx+fy)}^{2}}={{c}^{2}}({{x}^{2}}+{{y}^{2}})\]
D) \[{{(gx+fy)}^{2}}=c({{x}^{2}}+{{y}^{2}})\]
Correct Answer: D
Solution :
Equation of pair of tangents is \[S{{S}_{1}}={{T}^{2}}\], where \[T=x{{x}_{1}}+y{{y}_{1}}+g(x+{{x}_{1}})+f(y+{{y}_{1}})+c\] \[\Rightarrow c({{x}^{2}}+{{y}^{2}}+2gx+2fy+c)={{(gx+fy+c)}^{2}}\] \[\Rightarrow c({{x}^{2}}+{{y}^{2}})={{(gx+fy)}^{2}}\].
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