If the points represented by complex numbers $z_1=a+ib,z_2=a^1+i{b}^1\text{ and }{z}_1-z_2$ are collinear, then

By definition, $z_1=x_1+i{y}_1;z_2=x_2+i{y}_2;z_3=x_3+i{y}_3$ are collinear, if

$\left|\begin{array}{l}{x}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{y}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ x_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{y}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ x_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{y}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}a& b& 1\\ a^{\mathrm{\prime }}& b^{\mathrm{\prime }}& 1\\ a-a^{\mathrm{\prime }}& b-b^{\mathrm{\prime }}& 1\end{array}\right|=0$

$\Rightarrow a{b}^{\mathrm{\prime }}=a^{\mathrm{\prime }}b$