Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Two planes are coplanar if
$\left|\begin{array}{ccc}{x}_2-x_1& y_2-y_1& z_2-z_1\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right|=0$
$\left|\begin{array}{ccc}1& -1& -1\\ 1& 1& -k\\ k& 2& 1\end{array}\right|=0$
Applying ${\mathrm{C}}_2\to {\mathrm{C}}_2+{\mathrm{C}}_1,{\mathrm{C}}_3\to {\mathrm{C}}_3+{\mathrm{C}}_1$
$\Rightarrow \left|\begin{array}{ccc}1& 0& 0\\ 1& 2& 1-k\\ k& k+2& 1+k\end{array}\right|=0$
$\Rightarrow 1[2+2k-(k+2)(1-k)]=0$
$\Rightarrow 2+2k-\left(-k^2-k+2\right)=0$
$k^2+3k=0\Rightarrow k(k+3)=0$