A) \[\frac{a}{b}=\frac{4}{5}\]
B) \[\frac{a}{b}=\frac{5}{4}\]
C) \[a+b-2c=0\]
D) \[2a=b+c\]
Correct Answer: B
Solution :
Given, \[{{r}_{1}}=2{{r}_{2}}=3{{r}_{3}}\] \[\therefore \] \[\frac{\Delta }{s-a}=\frac{2\Delta }{s-b}=\frac{3\Delta }{s-c}\] \[\Rightarrow \] \[(s-b)=2(s-a)\] and \[(s-c)=3(s-a)\] \[\Rightarrow \] \[\left( \frac{a+b+c}{2}-b \right)=2\left( \frac{a+b+c}{2}-a \right)\] and \[\left( \frac{a+b+c}{2}-c \right)=3\left( \frac{a+b+c}{2}-a \right)\] \[\Rightarrow \] \[a+c-b=2(-a+b+c)\] and \[a+b-c=3(-a+b+c)\] \[\Rightarrow \] \[3a=3b+c\] and \[2a=b+2c\] \[\Rightarrow \] \[4a=5b\] \[\Rightarrow \] \[\frac{a}{b}=\frac{5}{4}\]
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