A) \[{{x}^{2}}+{{y}^{2}}+2x-6y+9=0\]
B) \[{{x}^{2}}+{{y}^{2}}-2x-4y+1=0\]
C) \[{{x}^{2}}+{{y}^{2}}+2x+4x+1=0\]
D) \[{{x}^{2}}+{{y}^{2}}-2x+4y+1=0\]
Correct Answer: A
Solution :
We have. \[\angle A=\frac{\pi }{2}\] \[\Rightarrow \] \[B{{C}^{2}}=C{{A}^{2}}+A{{B}^{2}}\] \[\Rightarrow \] \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\] \[\Rightarrow \] \[{{a}^{2}}=16+9\] \[\Rightarrow \] \[a=5\] \[\therefore \] \[R=\frac{a}{\sin A}=\frac{5}{2\sin \frac{\pi }{2}}=\frac{5}{2}\] \[Also,\]\[\Delta =\frac{1}{2}bc\sin A=\frac{1}{2}\times 4\times 3\sin \frac{\pi }{2}=6\] And, \[s=\frac{a+b+c}{2}=\frac{5+4+3}{2}=6\] Now, \[r=\frac{\Delta }{s}=\frac{6}{6}=1\] \[\therefore \] \[\frac{R}{r}=\frac{5}{2}\]
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