question_answer

A line through (0,0) cuts the circle \[{{x}^{2}}+{{y}^{2}}-2ax=0\] at A and B, then locus of the centre of the circle drawn on AB as a diameter is [RPET 2002]

A)            \[{{x}^{2}}+{{y}^{2}}-2ay=0\]                                        

B)            \[{{x}^{2}}+{{y}^{2}}+ay=0\]

C)            \[{{x}^{2}}+{{y}^{2}}+ax=0\]                                        

D)            \[{{x}^{2}}+{{y}^{2}}-ax=0\]

Correct Answer: D

Solution :

           Let chord AB is \[y=mx\]                     .....(i)                      Equation of CM, \[x+my=\lambda \]                    It is passing through (a, 0)                    \[\therefore \]  \[x+my=a\]                                                      .....(ii) From (i) and (ii), \[x+y.\,\frac{y}{x}=a\]Þ \[{{x}^{2}}+{{y}^{2}}=ax\]                                 \[\Rightarrow \]\[{{x}^{2}}+{{y}^{2}}-ax=0\] is the locus of the centre of the circle.