Question

If the system of linear equations
$(\cos \theta) x+(\sin \theta) y+\cos \theta=0 $
$(\sin \theta) x+(\cos \theta) y+\sin \theta=0 $
$(\cos \theta) x+(\sin \theta) y-\cos \theta=0$
is consistent, then find the number of possible values of $\theta \in[0,2 \pi]$.

Answer 1

Put $\Delta=0$
$\Rightarrow\begin{vmatrix}\cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & -\cos \theta\end{vmatrix}=0$
$\Rightarrow \cos \theta \cdot \cos 2 \theta=0 $
$\therefore \theta=\frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}$
But $\theta=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ (rejected)
As lines are parallel. So, system is inconsistent.