Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ internally. Find the locus of the centre of of $\Gamma$.
Denote $(a,0), (b,0)$ by $A, B$ respectively and let the center of $\Gamma$ be at $O$, and let its radius be $r$. Then $OA=a+r$ and $OB=b-r$, so $OA+OB=a+b$. Now, I am stuck here. I was thinking of using tangents and normals of the circles, but this would complicate the calculation. Please help.