Question

The lines $r\cos (\theta -\alpha )=a$, $r\cos (\theta -\beta )=b$ and $r\cos (\theta -\gamma )=c$ are concurrent if

• A. $\sum a\cos (\beta -\gamma )=0$
• B. $\sum a\cos (\beta +\gamma )=0$
• C. $\sum a\sin (\beta -\gamma )=0$
• D. $\sum a\sin (\beta +\gamma )=0$

Answer 1

The correct option is C $\sum a\sin (\beta -\gamma )=0$

Given equations are
$r\cos (\theta -\alpha )=a$
$\Rightarrow r\cos \alpha \cos \theta +r\sin \alpha \sin \theta -a=0$ ....(i)
$r\cos (\theta -\beta )=b$
$\Rightarrow r\cos \beta \cos \theta +r\sin \beta \sin \theta -b=0$ ....(ii)
$r\cos (\theta -\gamma )=c$
$\Rightarrow r\cos \gamma \cos \theta +r\sin \gamma \sin \theta -c=0$ ....(iii)
Since, these are concurrent
$\left|\begin{array}{ccc}\cos \alpha & \sin \alpha & -a\\ \cos \beta & \sin \beta & -b\\ \cos \gamma & \sin \gamma & -c\end{array}\right|=0$

$\Rightarrow a(\cos \beta \sin \gamma -\sin \beta \cos \gamma )+b(\sin \alpha \cos \gamma -\cos \alpha \sin \gamma )+c(\cos \alpha \sin \beta -\sin \alpha \cos \beta )=0$
$\Rightarrow \sum a\sin (\beta -\gamma )=0$