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 $\Gamma$.

My attempt:
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.


As sum of distance from $2$ points is constant, the locus is an ellipse. You can just write $$|z-B|+|z-A|=a+b$$

$z$ represents the locus you want to find out and $|x-y|$ represents distance between points $x$ and $y$. Just apply distance formula.


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