Find the locus of the centre of circles touching the y-axis and circle $x^2+y^2-2x=0$ externally.
My Attempt:
Circle touching y-axis is $(x-h)^2+(y-k)^2=h^2$.
If this is to touch the given circle externally then the distance between their centres should be equal to the sum of their radii.
$$\sqrt{(h-1)^2+(k-0)^2}=h+1\\\implies h^2+1-2h+k^2=h^2+1+2h\\\implies k^2=4h$$
Thus, the locus is a right facing parabola passing through origin. $y^2=4x$
But don't the circles $x^2+y^2+px=0, p\gt0$ also satisfy the given conditions? i.e. they touch y-axis and touch the given circle externally.
So, shouldn't negative x-axis also be the locus?
Or, is it required that the every point of the given circle must be touched?