# Locus of the circles touching another circle

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.

• I think that the locus is again a circle of some certain radius $\sqrt {a^2+b^2+r^2 - s^2}$ where $(a,b)$ is the centre of the given circle with radius $s$ and $r$ is the given radius of the circles whose locus of centres has to be determined. – Dbchatto67 Mar 20 at 5:14

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.