Question

In a $\Delta ABC$ if $\tan \left(A/2\right)=\frac{5}{6}$ and $\tan \left(B/2\right)=\frac{20}{37}$ the sides $a,b$ and $c$ are in

• A. $A.P$
• B. $G.P$
• C. $H.P$
• D. $noneofthese$

Answer 1

Answer: (A)

$A.P$

$\tan \frac{A}{2}=\frac{5}{6}$ and $\tan \frac{B}{2}=\frac{20}{31}$

We know that $s=\frac{a+b+c}{2}$ .....$\left(1\right)$

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

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

and $\tan \left(\frac{C}{2}\right)=\frac{\sqrt{(s-a)(s-b)}}{\sqrt{s(s-c)}}$ .....$\left(4\right)$

$\tan \frac{A}{2}\tan \left(\frac{B}{2}\right)=\frac{\sqrt{(s-b)(s-c)}}{\sqrt{s(s-a)}}\times \frac{\sqrt{(s-a)(s-c)}}{\sqrt{s(s-b)}}$

$=\frac{(s-c)}{s}$

$=\frac{\frac{a+b+c}{2}-c}{\frac{a+b+c}{2}}$ using $\left(1\right)$

$\Rightarrow \frac{5}{6}\times \frac{20}{37}=\frac{a+b+c}{a+b+c}$

$\Rightarrow 50a+50b+50c=111a+111b-111c$

$\Rightarrow 61a+61b-161c=0$ .......$\left(5\right)$

Similarly, $\frac{\tan \left(\frac{A}{2}\right)}{\tan \left(\frac{B}{2}\right)}$

$=\frac{\frac{\sqrt{(s-b)(s-c)}}{\sqrt{s(s-a)}}}{\frac{\sqrt{(s-a)(s-c)}}{\sqrt{s(s-b)}}}$

$\Rightarrow \frac{5}{6}\times \frac{37}{20}=\frac{s-b}{s-a}$

$\Rightarrow \frac{37}{24}=\frac{\frac{a+b+c}{2}-b}{\frac{a+b+c}{2}-a}$ using $\left(1\right)$

$\Rightarrow \frac{37}{24}=\frac{a-b+c}{-a+b+c}$

$\Rightarrow -37a+37b+37c=24a-24b+24c$

$\Rightarrow 61a-13c-61b=0$ .......$\left(6\right)$

Solving $\left(5\right)$ and $\left(6\right)$ we get

$61a-161c+61b=0$

$61a-13c-61b=0$ gives

$122a-174c=0$

$a=\frac{87c}{61}$

$\frac{a}{87}=\frac{c}{61}$ ........$\left(7\right)$

eqn$\left(5\right)-$eqn$\left(6\right)$ we get

$61a-161c+61b-61a+13c+61b=0$

$\Rightarrow b=\frac{74c}{61}$

$\Rightarrow \frac{b}{74}=\frac{c}{61}$ ........$\left(8\right)$

From $\left(7\right)$ and $\left(8\right)$ we get

$\frac{a}{87}=\frac{c}{61}=\frac{c}{61}$

Now, $a+c=(\frac{87}{74}+\frac{61}{74})b=2b$

Hence, $a+c=2b$

$\therefore a,b,c$ are in A.P