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Prove that the locus of centre of the ci...

Prove that the locus of centre of the circle which toches two given disjoint circles externally is hyperbola.

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As shown in the figure, variable circle S with centre C and radius r touches two given disjoint circles `S_(1)` and `S_(2)` having centres `C_(1)` and `C_(2)` and radii `r_(1)` and `r_(2)`, respectively.

Clearly, `"CC"_(1)=r+r_(1) and "CC"_(2)=r+r_(2)`
`therefore" CC"_(1)-"CC"_(2)=r_(1)-r_(2)(="constant")`
Thus, locus of centre C is hyperbola having foci `C_(1) and C_(2)`.
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