Question

In a $\Delta$ ABC, a, c, A are given and $b_1,b_2$ are two values of the third side b such that $b_2,2b_1$ , then sin A is equal to

• A. $\sqrt{\frac{9a^2-c^2}{8c^2}}$
• B. $\sqrt{\frac{9a^2-c^2}{8c^2}}$
• C. $\sqrt{\frac{9a^2+c^2}{8c^2}}$
• D. None of these

Answer 1

The correct option is B

$\sqrt{\frac{9a^2-c^2}{8c^2}}$

We have, $cosA=\frac{b^2+c^2-a^2}{2bc}\Rightarrow b^2-2bccosA+(c^2-a^2)=0\text{It is given that}{b}_{1}and{b}_2\text{are the roots of this equation.}\therefore ,b_1+b_2=2ccosAand{b}_1{b}_2=c^2-a^2\Rightarrow 3b_1=2ccosAand2b_1^2=c^2-a^2[\because b_2=2b_1]\Rightarrow 2{\left(\frac{2c}{3}cosA\right)}^2=c^2-a^2\Rightarrow 8c^2(1-si{n}^{2}A)=9c^2-9a^2\Rightarrow sinA=\sqrt{\frac{9a^2-c^2}{8c^2}}$