question_answer

The locus of the centres of the circles, which touch the circle, \[{{x}^{2}}+{{y}^{2}}=1\]externally, also touch the y-axis and lie in the first quadrant, is : [JEE Main 10-4-2019 Afternoon]

A) \[y=\sqrt{1+4x},x\ge 0\]        

B) \[x=\sqrt{1+4y},y\ge 0\]

C) \[x=\sqrt{1+2y},y\ge 0\]

D)   \[y=\sqrt{1+2x},x\ge 0\]

Correct Answer: D

Solution :

          \[\sqrt{{{h}^{2}}+{{k}^{2}}}=|h|+1\] \[\Rightarrow {{x}^{2}}+{{y}^{2}}={{x}^{2}}+1+2x\] \[\Rightarrow {{y}^{2}}=1+2x\] \[\Rightarrow y=\sqrt{1+2x};x\ge 0.\]