Question

Consider a $△ABC$ whose sides $a,b$ and $c$ are such that $a^2,b,c^2$ are in $G.P.,$ then which of the following statement(s) is/are correct ?
(where $a,b,c$ are sides of $△ABC$ opposite to $\angle A,\angle B$ and $\angle C$ respectively)

• A. $\sin A=b\sin C$
• B. $\sin B=c\sin A$
• C. $\sin C=a\sin B$
• D. $\sin B=a\sin C$

Answer 1

Answer: (B), (D)

$\sin B=a\sin C$

Given : $a^2,b,c^2$ are in $G.P.$
$\Rightarrow b^2=a^2{c}^2\Rightarrow b=ac\Rightarrow \frac{b}{c}=a\cdots \left(i\right)$
or $\frac{b}{a}=c\cdots \left(ii\right)$
From sine rule, we have
$\frac{b}{c}=\frac{\sin B}{\sin C}$
and $\frac{b}{a}=\frac{\sin B}{\sin A}$
Putting the above values in $\left(i\right)$ and $\left(ii\right),$ we get
$\sin B=a\sin C$ or $\sin B=c\sin A$