The locus of the centre of circles, which touch the circle $x^2+y^2=1$ , externally also touch the Y-axis and lie in the first quadrant is

$\begin{array}{l}C=\left(x_1,y_1\right) be Locus point ,centreO=(0,0)\\ Circles touch externally\Rightarrow C_1{C}_2=r_1+r_2\\ CO=1+r\\ circle touch yaxis \Rightarrow r=\left|x_1\right|\\ \sqrt{x_1^2+y_1^2}=1+\left|x_1\right|=1+x_1\Rightarrow y_1^2=1+2x_1\\ y=\sqrt{1+2x}(x\ge 0)\end{array}$