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Q.
The number of common tangents of the circles $ {{x}^{2}}+{{y}^{2}}-2x-1=0 $ and $ {{x}^{2}}+{{y}^{2}}-2y-7=0 $ is
Jharkhand CECEJharkhand CECE 2011
Solution:
Let $ {{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}-2x-1=0 $
Centre, $ {{C}_{1}}=(1,\,\,0) $ and Radius
$ {{r}_{1}}=\sqrt{1+0+1}=\sqrt{2} $ and $ {{S}_{2}}={{x}^{2}}+{{y}^{2}}-2y-7=0 $
Centre, $ {{C}_{2}}(0,\,\,1) $ and Radius
$ {{r}_{2}}=\sqrt{0+1+7}=2\sqrt{2} $
Here, $ {{C}_{1}}{{C}_{2}}=\sqrt{{{(1-0)}^{2}}+{{(0-1)}^{2}}}=\sqrt{2} $
$ \because $ $ {{C}_{1}}{{C}_{2}}={{r}_{2}}-{{r}_{1}} $
$ \therefore $ Circles touch each other internally and so only one common tangent can be drawn to given two circles.