Q67 / 160 · TS EAMCET 2019 (03 May Shift 1) · Mathematics

In any triangle $A B C$, if $a: b: c=2: 3: 4$, then $R: r=$

  • A
    $8: 3$
  • B
    $16: 9$
  • C
    $5: 16$
  • Correct answer
    D
    $16: 5$

Solution

We have, Let $$ a: b: c=2: 3: 4 $$ $$ \begin{aligned} & \therefore \quad R=\frac{a b c}{4 \Delta} \text { and } r=\frac{\Delta}{s} \\ & \therefore \quad \frac{R}{r}=\frac{s \cdot a b c}{4 \Delta^2} \\ & \Rightarrow \quad \frac{R}{r}=\frac{s \cdot(2 k)(3 k)(4 k)}{4 s(s-2 k)(s-3 k)(s-4 k)} \\ & \Rightarrow \quad \frac{R}{r}=\frac{6 k^3}{\left(\frac{9 k}{2}-2 k\right)\left(\frac{9 k}{2}-3 k\right)\left(\frac{9 k}{2}-4 k\right)} \\ & {\left[\because s=\frac{a+b+c}{2}\right]} \\ & \end{aligned} $$ $$ \begin{array}{ll} \Rightarrow & \frac{R}{r}=\frac{6 \cdot 2^3 \cdot k^3}{5 k \cdot 3 k \cdot k} \Rightarrow \frac{R}{r}=\frac{16}{5} \\ \therefore & R: r=16: 5 \end{array} $$

Hence, the correct answer is option: 'D'.

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