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Definition of Sets

In a Δ ABC,...

Question

In a $Δ A B C,$ the perimeter is $2 s$ and the exradii are $r_{1}, r_{2}$ and $r_{3} .$ Then $r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}$ is equal to?

s^{2}

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2 s^{2}

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3 s^{2}

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4 s^{2}

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Solution

The correct option is A $s^{2}$
Consider, $t = r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}$
$\Rightarrow t = \frac{△^{2}}{(s - a) (s - b)} + \frac{△^{2}}{(s - b) (s - c)} + \frac{△^{2}}{(s - c) (s - a)}$
$\Rightarrow t = △^{2} (\frac{s - c + s - a + s - b}{(s - a) (s - b) (s - c)}) = △^{2} [\frac{3 s - (a + b + c)}{(s - a) (s - b) (s - c)}]$
$\Rightarrow t = \frac{s (s - a) (s - b) (s - c) (3 s - 2 s)}{(s - a) (s - b) (s - c)} = s^{2}$
Ans: A

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Similar questions

r_{1} (r_{2} + r_{3}) / a = r_{2} (r_{3} + r_{1}) / b = r_{3} (r_{1} + r_{2}) / c

- 1

Δ A B C

a < b < c

r

r_{1}, r_{2}, r_{3}

A, B, C

r < r_{1} < r_{2} < r_{3}

- 2

Δ A B C

r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = \frac{r_{1} r_{2} r_{3}}{r}

△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}

r_{1}, r_{2}

r_{3}

2 s

Δ A B C

(1 - \frac{r_{1}}{r_{2}}) (1 - \frac{r_{1}}{r_{3}}) = 2

Δ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = 2 r^{2}

\frac{r_{1} (r_{2} + r_{3})}{\sqrt{r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}}} =

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