The equation of the plane, passing through the point \((-1,2,-3)\) and parallel to the lines. \(\frac{x-1}{3}=\frac{y-2}{2}=\frac{z}{-4}\) and \(\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}\), is

Let \(\left(x_1, y_1, z_1\right)=(-1,2,-3)\)
\(a_1, b_1, c_1=3,2,-4\) and
\(a_2, b_2, c_2=2,-3,2\)
\(\therefore \quad\) The equation of required plane is
\(\begin{aligned} & \left|\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\ \mathrm{a}_1 & \mathrm{~b}_1 & \mathrm{c}_1 \\ \mathrm{a}_2 & \mathrm{~b}_2 & \mathrm{c}_2\end{array}\right|=0 \\ & \Rightarrow\left|\begin{array}{ccc}x+1 & y-2 & \mathrm{z}+3 \\ 3 & 2 & -4 \\ 2 & -3 & 2\end{array}\right|=0\end{aligned}\)
\(\begin{aligned} & \Rightarrow(x+1)(-8)-(y-2)(14)+(z+3)(-13)=0 \\ & \Rightarrow-8 x-8-14 y+28-13 z-39=0 \\ & \Rightarrow-8 x-14 y-13 z-19=0 \\ & \Rightarrow 8 x+14 y+13 z+19=0\end{aligned}\)