Question

Statement 1: In a $\Delta ABC$, if $b+c=3a$, then $\cot \frac{B}{2}\cot \frac{C}{2}=2$

Statement 2: In a $\Delta ABC$, if $\frac{b+c}{a}=\frac{m}{n}(m>n$ and $m,n$ are positive)

then $\cot \frac{B}{2}\cot \frac{C}{2}=\frac{m+n}{m-n}$

• A. Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement- 1
• B. Statement-1 is true, Statement-2 is true, Statement-2 is not correct explanation for Statement- 1
• C. Statement-1 is true, Statement-2 is false
• D. Statement-1 is false, Statement-2 is true

Answer 1

Answer: (A)

Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement- 1

we know that , $tan\frac{B}{2}=\frac{△}{s(s-b)}tan\frac{C}{2}=\frac{△}{s(s-c)}$

So, $cot\frac{B}{2}cot\frac{C}{2}=\frac{s^2(s-b)(s-c)}{{△}^2}=\frac{s^2(s-a)(s-b)(s-c)}{{△}^2(s-a)}=\frac{s}{s-a}$ $(as{△}^2=s(s-a\left)\right(s-b\left)\right(s-c\left)\right)$

Now, it is given that $\frac{b+c}{a}=\frac{m}{n}\Rightarrow b+c=\frac{m}{n}a$

we can write $\frac{s}{s-a}=\frac{\frac{a+b+c}{2}}{\frac{b+c-a}{2}}=\frac{a+b+c}{b+c-a}=\frac{\frac{(m+n)a}{n}}{\frac{(m-n)a}{n}}=\frac{m+n}{m-n}$

Now, we can see that both statements are correct and Statement-2 is the correct explaination of statement -1

Therefore, Correct Answer is $A$