A) \[{{x}^{2}}+{{y}^{2}}+3x-6y+5=0\]
B) \[{{x}^{2}}+{{y}^{2}}+3x-6y-31=0\]
C) \[{{x}^{2}}+{{y}^{2}}+3x-6y+\frac{29}{4}=0\]
D) None of these
Correct Answer: B
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 \[{{x}^{2}}+{{y}^{2}}+3x-6y-31=0\]
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