question_answer

The intercept on the line \[y=x\] by the circle \[{{x}^{2}}+{{y}^{2}}-2x=0\] is AB, equation of the circle on AB as a diameter is [AIEEE 2004]

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: D

Solution :

           Given, circle is \[{{x}^{2}}+{{y}^{2}}-2x=0\]                                  ......(i)                    and line is \[y=x\]                                             .....(ii)                                                        Putting \[y=x\] in (i),                    We get \[2{{x}^{2}}-2x=0\Rightarrow x=0,\,1\]                    From (i), \[y=0\], 1                                Let \[A=(0,\,0)\], \[B=(1,\,1)\]                    Equation of required circle is                    \[(x-0)\,(x-1)+(y-0)\,(y-1)=0\]                     or \[{{x}^{2}}+{{y}^{2}}-x-y=0\].