Question

System of equations
$x+2y+z=0,2x+3y-z=0$ and $\left(\tan \theta \right)x+y-3z=0$ has non-trivial solution, then number of values(s) of $\theta \in (-\pi ,\pi )$ is equal to

Answer 1

For non-trivial solutions
$\left|\begin{array}{ccc}1& 2& 1\\ 2& 3& -1\\ \tan \theta & 1& -3\end{array}\right|=0$

$R_1\to R_1+R_2$
$\left|\begin{array}{ccc}3& 5& 0\\ 2& 3& -1\\ \tan \theta & 1& -3\end{array}\right|=0$

$R_2\to 3R_2-R_3$
$\left|\begin{array}{ccc}3& 5& 0\\ 6-\tan \theta & 8& 0\\ \tan \theta & 1& -3\end{array}\right|=0$

$\Rightarrow -3(24-(30-5\tan \theta \left)\right)=0$
$\Rightarrow \tan \theta =\frac{6}{5}$
Hence, the number of solutions in $(-\pi ,\pi )$ is $2$