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Q.
The locus of the centre of a circle of radius 2 which rolls on the outside of the circle $ {{x}^{2}}+{{y}^{2}}+3x-6y-9=0 $ is
J & K CETJ & K CET 2003
Solution:
The centre and radius of circle $ {{x}^{2}}+{{y}^{2}}+3x-6y-9=0 $ are $ {{C}_{1}}\left( -\frac{3}{2},3 \right) $ and $ {{r}_{1}}=\frac{9}{2} $
Let the centre and radius of required circle are $ {{C}_{2}}(g,f) $ and $ {{r}_{2}}=2 $ .
Since, the required circle is rolled outside the given circle.
$ \therefore $ $ {{C}_{1}}\,\,{{C}_{2}}={{r}_{1}}+{{r}_{2}} $
$ \Rightarrow $ $ \sqrt{{{\left( g+\frac{3}{2} \right)}^{2}}+{{(f-3)}^{2}}}=2+\frac{9}{2} $
$ \Rightarrow $ $ {{g}^{2}}+\frac{9}{4}+3g+{{f}^{2}}+9-6f=31 $ Hence, locus of the centre is $ {{x}^{2}}+{{y}^{2}}+3x-6y-31=0 $