Question

Assertion :In any triangle ABC the minimum value of $\frac{r_1+r_2+r_3}{r}$ is 9 Reason: For any three numbers $AM\ge GM$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

AM of $r_1,r_2,r_3$ is $\frac{r_1+r_2+r_3}{3}$ ....(1)
HM of $r_1,r_2,r_3$ is $\frac{3}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}}$ ....(2)
Now, $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}$
Using $r_1=\frac{\Delta }{(s-a)},r_2=\frac{\Delta }{(s-b)},r_3=\frac{\Delta }{(s-c)}$
$=\frac{3s-2s}{\Delta }=\frac{1}{r}$
So, HM of $r_1,r_2,r_3$ is $\frac{3}{r}$ ....(3)
AM $\ge$ HM
$\frac{r_1+r_2+r_3}{3}\ge \frac{3}{r}$
$\Rightarrow \frac{r_1+r_2+r_3}{r}\ge 9$
Hence, assertion is correct but reason is incorrect.