If $3 x^{2}+x y-y^{2}-3 x+6 y+k=0$ represents a pair of lines, then
$$
\begin{aligned}
\left|\begin{array}{ccc}
3 & \frac{1}{2} & \frac{-3}{2} \\
\frac{1}{2} & -1 & 3 \\
\frac{-3}{2} & 3 & k
\end{array}\right|=0 \\
\Rightarrow 3(-k-9)-\frac{1}{2}\left(\frac{1}{2} k+\frac{9}{2}\right)-\frac{3}{2}\left(\frac{3}{2}-\frac{3}{2}\right) &=0 \\
\Rightarrow \quad-3 k-27-\frac{1}{4} k-\frac{9}{4} &=0 \\
\Rightarrow \quad-\frac{13}{4} k-\frac{117}{4} &=0 \\
\Rightarrow &=-9
\end{aligned}
$$