Assertion -In any ABC the minimum value of r1-r2-r3r is 9 Reason- A-M -G-M

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Standard XII

Mathematics

Greatest Integer Function

Assertion :In...

Question

Assertion :In any $△ A B C$ the minimum value of $\frac{r_{1} + r_{2} + r_{3}}{r}$ is $9$ Reason: A.M $\geq$ G.M

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 B Both Assertion and Reason are individually correct but Reason is not the correct explanation of Assertion
$\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}}$
$= \frac{s - a}{△} + \frac{s - b}{△} + \frac{s - c}{△}$
$= \frac{3 s - (a + b + c)}{△}$
$= \frac{3 s - 2 s}{△}$
$= \frac{s}{△} = \frac{1}{r}$
$∵$ A.M $\geq$ H.M
$∴ \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} \geq \frac{3}{(\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}})} = \frac{3}{\frac{1}{r}} = 3 r$
or $\frac{r_{1} + r_{2} + r_{3}}{r} \geq 9$
$∴$ Minimum value of $\frac{r_{1} + r_{2} + r_{3}}{r}$ is $9$

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Assertion :In any △ABC the minimum value of r1+r2+r3r is 9 Reason: A.M≥G.M

Assertion :In any $△ A B C$ the minimum value of $\frac{r_{1} + r_{2} + r_{3}}{r}$ is $9$ Reason: A.M $\geq$ G.M