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Existence of Limit

In Δ ABC ...

Question

In $Δ$ $A B C$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively.
If $a : b : c = 3 : 7 : 8$ , then $R : r$ is equal to ?

2 : 7

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7 : 2

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3 : 7

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7 : 3

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Solution

The correct option is B $7 : 2$
$a : b : c = 3 : 7 : 8$
Let, $a = 3 k$ , $b = 7 k$ , $c = 8 k$
$R = \frac{a b c}{4 Δ}$
$r = \frac{Δ}{s}$

Now, $\frac{R}{r} = \frac{\frac{a b c}{4 Δ}}{\frac{Δ}{s}}$
$\frac{R}{r} = \frac{a b c . s}{4 Δ^{2}}$
$\frac{R}{r} = \frac{a b c . s}{4 s (s - a) (s - b) (s - c)}$
$\frac{R}{r} = \frac{a b c}{4 (s - a) (s - b) (s - c)}$

We know, $s = \frac{a + b + c}{2} = \frac{3 k + 7 k + 8 k}{2} = 9 k$
Thus, $\frac{R}{r} = \frac{3 k . 7 k . 8 k}{4 (6 k) (2 k) (k)}$
$\frac{R}{r} = \frac{7}{2}$

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