Locus of the circles touching another circle

Question

Find the locus of centre of all circles which are of given radius and touch a given circle. I can make out that it is a circle but I am unable to prove it.

Answer 1

Consider a given circle $S^1$ have center (a,b) and radius $r_1$. Let the varying circle with center $(x, y)$ and constant radius $r_2$touches $S^1$. Since the circles touches each other the distance between their centers is constant , $r_1+r_2$.

Thus equating the distance gives $$(x-a)^2+( y-b)^2=(r_1+r_2)^2.$$Which is an equation of a circle centered at $(a,b)$ and radius $(r_1+r_2)$. Thus the locus $(x,y)$ forms a circle.