The locus of the centres of the circles,

Question:

The locus of the centres of the circles, which touch the circle, $x^{2}+y^{2}=1$ externally, also touch the $\mathrm{y}$-axis and lie in the first quadrant, is :

  1. $y=\sqrt{1+4 x}, x \geq 0$

  2. $x=\sqrt{1+4 y}, y \geq 0$

  3. $x=\sqrt{1+2 y}, y \geq 0$

  4. $\mathrm{y}=\sqrt{1+2 \mathrm{x}}, \mathrm{x} \geq 0$


Correct Option: , 4

Solution:

$\sqrt{\mathrm{h}^{2}+\mathrm{k}^{2}}=|\mathrm{h}|+1$

$\Rightarrow x^{2}+y^{2}=x^{2}+1+2 x$

$\Rightarrow y^{2}=1+2 x$

$\Rightarrow y=\sqrt{1+2 x} ; \quad x \geq 0$

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