Question

If $m=4,n=5$ and $k\cot \theta =m\cot A-n\cot B$, then find the value of $k$.

• A. $6$
• B. $1$
• C. $9$
• D. $3$

Answer 1

Answer: (C)

$9$

Given $m=4,n=5$

In triangle $PSQ$, by sine rule we get $\frac{QS}{\sin \left(QPS\right)}=\frac{PS}{\sin \left(PQS\right)}$

$\Rightarrow \frac{PS}{QS}=\frac{\sin \left(PQS\right)}{\sin \left(QPS\right)}=\frac{\sin (\theta -A)}{\sin \left(A\right)}=\sin \theta \cot A-\cos \theta$

In triangle $PSR$, by \sine rule we get $\frac{RS}{\sin \left(RPS\right)}=\frac{PS}{\sin \left(PRS\right)}$

$\Rightarrow \frac{PS}{RS}=\frac{\sin \left(PRS\right)}{\sin \left(RPS\right)}=\frac{\sin (\theta +B)}{\sin \left(B\right)}=\sin \theta \cot B+\cos \theta$

Now divide above two equations, we get $\frac{RS}{QS}=\frac{\sin \theta \cot A-\cos \theta }{\sin \theta \cot B+\cos \theta }$

$\Rightarrow \frac{n}{m}=\frac{\cot A-\cot \theta }{\cot B+\cot \theta }$

$\Rightarrow m\cot A-n\cot B=(n+m)\cot \theta =k\cot \theta$

Therefore, the value of $k=m+n=4+5=9$

Therefore, option $C$ is correct.