Search any question & find its solution
Question:
Answered & Verified by Expert
In a $\triangle A B C, \frac{a}{b^2-c^2}+\frac{c}{b^2-a^2}=0$, then $B$ is equal to
Options:
Solution:
1349 Upvotes
Verified Answer
The correct answer is:
$\frac{\pi}{3}$
We have,
$\begin{aligned} & \frac{a}{b^2-c^2}+\frac{c}{b^2-a^2}=0 \\ & \Rightarrow \quad \frac{2 R \sin A}{4 R^2\left(\sin ^2 B-\sin ^2 C\right)} \\ & \quad+\frac{2 R \sin C}{4 R^2\left(\sin ^2 B-\sin ^2 A\right)}=0\end{aligned}$
$\begin{aligned} & \Rightarrow \quad \frac{1}{2 R} \cdot \frac{\sin A}{\sin (B+C) \sin (B-C)} \\ & +\frac{1}{2 R} \cdot \frac{\sin C}{\sin (B+A) \sin (B-A)}=0 \\ & \Rightarrow \quad \frac{1}{2 R} \cdot \frac{\sin A}{\sin A-\sin (B-C)} \\ & +\frac{1}{2 R} \cdot \frac{\sin C}{\sin C \sin (B-A)}=0 \\ & \rightarrow \quad \frac{1}{2 R \sin (R-C)}+\frac{1}{2 R \sin (B-A)}=0 \\ & \Rightarrow \quad \sin (B-C)+\sin (B-A)=0 \\ & \Rightarrow \quad 2 \sin \frac{2 B-A-C}{2} \cos \frac{-C+A}{2}=0 \\ & \Rightarrow \quad \sin \frac{2 B-A-C}{2}=0 \\ & \Rightarrow \quad 2 l l=A+C \\ & 2 R=180^n-B \\ & 3 b=180^{\circ} \\ & B=\frac{\pi}{3} \\ & \end{aligned}$
$\begin{aligned} & \frac{a}{b^2-c^2}+\frac{c}{b^2-a^2}=0 \\ & \Rightarrow \quad \frac{2 R \sin A}{4 R^2\left(\sin ^2 B-\sin ^2 C\right)} \\ & \quad+\frac{2 R \sin C}{4 R^2\left(\sin ^2 B-\sin ^2 A\right)}=0\end{aligned}$
$\begin{aligned} & \Rightarrow \quad \frac{1}{2 R} \cdot \frac{\sin A}{\sin (B+C) \sin (B-C)} \\ & +\frac{1}{2 R} \cdot \frac{\sin C}{\sin (B+A) \sin (B-A)}=0 \\ & \Rightarrow \quad \frac{1}{2 R} \cdot \frac{\sin A}{\sin A-\sin (B-C)} \\ & +\frac{1}{2 R} \cdot \frac{\sin C}{\sin C \sin (B-A)}=0 \\ & \rightarrow \quad \frac{1}{2 R \sin (R-C)}+\frac{1}{2 R \sin (B-A)}=0 \\ & \Rightarrow \quad \sin (B-C)+\sin (B-A)=0 \\ & \Rightarrow \quad 2 \sin \frac{2 B-A-C}{2} \cos \frac{-C+A}{2}=0 \\ & \Rightarrow \quad \sin \frac{2 B-A-C}{2}=0 \\ & \Rightarrow \quad 2 l l=A+C \\ & 2 R=180^n-B \\ & 3 b=180^{\circ} \\ & B=\frac{\pi}{3} \\ & \end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.