A) \[{{x}^{2}}+{{y}^{2}}-3x=0\]
B) \[{{x}^{2}}+{{y}^{2}}-3y=0\]
C) \[{{x}^{2}}+{{y}^{2}}-x=0\]
D) \[{{x}^{2}}+{{y}^{2}}-y=0\]
Correct Answer: C
Solution :
The given circle is\[{{x}^{2}}+{{y}^{2}}-2x=0\]. Let \[({{x}_{1}},\ {{y}_{1}})\] be the middle point of any chord of this circle, than its equation is \[{{S}_{1}}=T\]. or \[x_{1}^{2}+y_{1}^{2}-2{{x}_{1}}=x{{x}_{1}}+y{{y}_{1}}-(x+{{x}_{1}})\] If it passes through (0, 0), then \[x_{1}^{2}+y_{1}^{2}-2{{x}_{1}}=-{{x}_{1}}\Rightarrow x_{1}^{2}+y_{1}^{2}-{{x}_{1}}=0\] Hence the required locus of the given point \[({{x}_{1}},\ {{y}_{1}})\] is \[{{x}^{2}}+{{y}^{2}}-x=0\].
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