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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.

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    $\begingroup$ 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. $\endgroup$
    – little o
    Mar 20, 2019 at 5:14

1 Answer 1

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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.

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