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.
Asked
Active
Viewed 81 times
0
-
2I 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. – little o Mar 20 '19 at 05:14
1 Answers
0
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.
walber97
- 151