question_answer

The locus of the middle points of those chords of the circle \[{{x}^{2}}+{{y}^{2}}=4\]which subtend a right angle at the origin is  [MP PET 1990; IIT 1984; RPET 1997; DCE 2000, 01]

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

B)            \[{{x}^{2}}+{{y}^{2}}=4\]

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

D)            \[{{(x-1)}^{2}}+{{(y-2)}^{2}}=5\]

Correct Answer: C

Solution :

           Let the mid-point of chord is (h,k). Also radius of circle is 2. Therefore                           \[\frac{OC}{OB}=\cos {{45}^{o}}\Rightarrow \frac{\sqrt{{{h}^{2}}+{{k}^{2}}}}{2}=\frac{1}{\sqrt{2}}\Rightarrow {{h}^{2}}+{{k}^{2}}=2\]                    Hence locus is\[{{x}^{2}}+{{y}^{2}}=2\].