Question

In $\Delta ABC$, with the usual notations, if $\left(tan\frac{A}{2}\right)\left(tan\frac{B}{2}\right)=\frac{3}{4}$ then $a+b=.......$

• A. $4c$
• B. $2c$
• C. $7c$
• D. $3c$

Answer 1

Answer: (C)

We have, In $\Delta ABC$
$\left(\tan \frac{A}{2}\right)\left(\tan \frac{B}{2}\right)=\frac{3}{4}$
$\Rightarrow \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}=\frac{3}{4}$
$\Rightarrow \sqrt{\frac{(s-b)(s-c)(s-a)(s-c)}{s(s-a)\cdot s(s-b)}}=\frac{3}{4}$
$\Rightarrow \frac{(s-c)}{s}=\frac{3}{4}$
$\Rightarrow \frac{\frac{a+b+c}{2}-c}{\frac{a+b+c}{2}}=\frac{3}{4}$
$\Rightarrow \frac{a+b-c}{a+b+c}=\frac{3}{4}$
$\Rightarrow 4a+4b-4c=3a+3b+3c$
$\Rightarrow a+b=7c$