Question

If tan$\left(\frac{A}{2}\right)=\frac{5}{6},tan\left(\frac{B}{2}\right)=\frac{20}{37}$ then a,b,c are in

• A. A.P
• B. G.P
• C. H.P
• D. NONE OF THESE.

Answer 1

The correct option is D NONE OF THESE.

We have,

$\tan \frac{A}{2}=\frac{5}{6},\tan \frac{B}{2}=\frac{20}{37}$

Then, we know that,

$\tan \frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}=\frac{5}{6}.......\left(1\right)$

$\tan \frac{A}{2}=\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}=\frac{20}{3}.......\left(2\right)$

By equation (1) multiply (2) to and we get.

$\sqrt{\frac{(s-b)(s-c)}{s(s-a)}\times \frac{(s-a)(s-c)}{s(s-b)}}=\frac{5}{6}\times \frac{20}{37}$

$\sqrt{{\left(\frac{s-c}{s}\right)}^2}=\frac{100}{111}$

$\frac{s-c}{s}=\frac{100}{111}$

$111s-111c=100s$

$111s-100s=111c$

$11s=111c$

$11\frac{(a+b+c)}{2}=111c$

$11a+11b+11c=222c$

$11a+11b=222c-11c$

$11a+11b=211c$

$c=\frac{11a+11b}{211}$

Hence, this is the answer.