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 :
Let (h, k) be the centre of the circle which rolls on the outside of the given circle. Centre of the given circle is \[\left( \frac{-3}{2},\ 3 \right)\] and its radius\[=\sqrt{\frac{9}{4}+9+9}=\frac{9}{2}\]. Clearly, (h, k) is always at a distance equal to the sum \[\left( \frac{9}{2}+2 \right)\]\[=\frac{13}{2}\] of the radii of two circles from\[\left( -\frac{3}{2},\ 3 \right)\]. Therefore \[{{\left( h+\frac{3}{2} \right)}^{2}}+{{(k-3)}^{2}}={{\left( \frac{13}{2} \right)}^{2}}\] \[\Rightarrow {{h}^{2}}+{{k}^{2}}+3h-6k+\frac{9}{4}+9-\frac{169}{4}=0\] \[\Rightarrow \]Hence locus of (h, k) is \[{{x}^{2}}+{{y}^{2}}+3x-6y-31=0\].
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