The locus of the centres of the circles, which touch the circle x^2+y^2=1 externally, also…

Question

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:

Accepted Answer

Let the centre be $(h,k)$, radius $=h$ (tangent to $y$-axis). External tangency to the unit circle:
$$\sqrt{h^2+k^2}=1+h\Rightarrow h^2+k^2=1+2h+h^2\Rightarrow k^2=1+2h.$$

$$y=\sqrt{1+2x},\ x\ge0.$$

Solutions · 1