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

Answer: 2

Put $\Delta =0$
$\Rightarrow \left|\begin{array}{ccc}\cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & -\cos \theta \end{array}\right|=0$
$\Rightarrow \cos \theta \cdot \cos 2\theta =0$
$\therefore \theta =\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\frac{5\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}$
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.