The equation of line joining $(1,2)\text{\hspace{0.17em}and\hspace{0.17em}}(3,6)$ using determinants is

Let $P(x,y)$ be any point on the line joining points  $A(1,2)\text{\hspace{0.17em}and\hspace{0.17em}}B(3,6)$.  Then, the points  $A,B\text{\hspace{0.17em}and\hspace{0.17em}}P$ are collinear.  Therefore the area of triangle  $ABP$ will be zero.

$\Delta =0$

$\frac{1}{2}\left|\begin{array}{ccc}1& 2& 1\\ 3& 6& 1\\ x& y& 1\end{array}\right|=0$

$\frac{1}{2}|1(6-y)-2(3-x)+1(3y-6x)|=0$

$2y-4x=0$

$y=2x$