Question

If in a triangle $\sin A:\sin C=\sin (A-B):(B-C)$ then $a^2:b^2:c^2$

• A. are in A.P.
• B. are in G.P.
• C. are in H.P.
• D. none of these

Answer 1

The correct option is A are in A.P.

In a triangle, we known sum of side =$\pi$

In $\Delta ABC$, $A+B+C=\pi$

$\Rightarrow A=\pi -(B+C)$

and $C=\pi -(A+B)$

Now $\frac{\sin A}{\sin C}=\frac{\sin (\pi -(B+C\left)\right)}{\sin (\pi -(A+B\left)\right)}=\frac{\sin (B+C)}{\sin (A+B)}=\frac{\sin (A-B)}{\sin (B-C)}$

$\Rightarrow \sin (B+C)\sin (B-C)=\sin (A-B)\sin (A+B)$

$\Rightarrow {\sin }^{2}B-{\sin }^{2}C={\sin }^{2}A-{\sin }^{2}B$ ......(1)

Now by since rule, we have

$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=K$

$\Rightarrow \sin A=ak,\sin B=bk$ and $\sin C=ck$

Substituting in (1)

$b^2{k}^2-c^2{k}^2=a^2{k}^2-b^2{k}^2$

$b^2-c^2=a^2-b^2$

$\Rightarrow 2b^2=c^2+a^2$

$\Rightarrow a^2,b^2$ and $c^2$ are in A.P