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 :
Key Idea: If a tangents drawn through a point \[({{x}_{1}},{{y}_{1}})\]circles S, then the equation of pair of tangents is \[S{{S}_{1}}={{T}^{2}}.\] Let \[S={{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] The tangents are drawn through the origin (0, 0) to the circle is \[S{{S}_{1}}={{T}^{2}}\] \[\Rightarrow \] \[({{x}^{2}}+{{y}^{2}}+2gx+2fy+c)c\] \[={{(gx+fy+c)}^{2}}\] \[\Rightarrow \] \[c({{x}^{2}}+{{y}^{2}})={{(gx+fy)}^{2}}\]
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