Assertion :In any triangle minimum value of $\frac{r_{1} r_{2} r_{3}}{r^{3}}$ is $27$ Reason: If $a_{1} + a_{2} + a_{3} + . . . + a_{n} = k$ where $k$ is a constant, then the value of $a_{1} a_{2} a_{3} . . . a_{n}$ is minimum when
$a_{1} = a_{2} = a_{3} = . . . = a_{n}$

Both Assertion and Reason are individually true and Reason is the correct explanation of Assertion.

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Both Assertion and Reason are individually correct but Reason is not the correct explanation of Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is C Assertion is correct but Reason is incorrect
$∵$ G.M $\geq$ A.M
${(r_{1} r_{2} r_{3})}^{\frac{1}{3}} \geq \frac{3}{\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}}}$
$= \frac{3}{\frac{1}{r}}$
$= 3 r$
$∴ {(r_{1} r_{2} r_{3})}^{\frac{1}{3}} \geq 3 r$
or $\frac{r_{1} r_{2} r_{3}}{r^{3}} \geq 27$
Also, if $a_{1} + a_{2} + a_{3} + . . . + a_{n} = k$ (constant)
Then, the value $a_{1} a_{2} a_{3} . . ., a_{n}$ is greatest.
when $a_{1} = a_{2} = a_{3} = . . . = a_{n}$

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