A) \[{{x}^{2}}+{{y}^{2}}-x-y=0\]
B) \[{{x}^{2}}+{{y}^{2}}-x+y=0\]
C) \[{{x}^{2}}+{{y}^{2}}+x+y=0\]
D) \[{{x}^{2}}+{{y}^{2}}+x-y=0\]
Correct Answer: A
Solution :
The equation of line is\[y=x\] ...(i) and equation of circle is \[{{x}^{2}}+{{y}^{2}}-2x=0\] ...(ii) On solving Eqs. (i) and (ii), we get \[\Rightarrow \] \[{{x}^{2}}+{{x}^{2}}-2x=0\] \[\Rightarrow \] \[2{{x}^{2}}-2x=0\] \[\Rightarrow \] \[2x(x-1)=0\] \[\Rightarrow \] \[x=0,\text{ }x=1\] when \[x=0,\text{ }y=0\] when \[x=1,\text{ }y=1\] Let co-ordinates of A are (0, 0) and co-ordinates of B are (1, 1). \[\therefore \]Equation of circle (AB as a diameter) is \[(x-{{x}_{1}})(x-{{x}_{2}})+(y-{{y}_{1}})(y-{{y}_{2}})=0\] \[\Rightarrow \] \[(x-0)(x-1)+(y-0)(y-1)=0\] \[\Rightarrow \] \[x(x-1)+y(y-1)=0\] \[\Rightarrow \] \[{{x}^{2}}-x+{{y}^{2}}-y=0\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}-x-y=0\]
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