Answer
See below.
Work Step by Step
We know that for a matrix
\[
\left[\begin{array}{rrr}
a & b & c \\
d &e & f \\
g &h & i \\
\end{array} \right]
\]
the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$
Hence here $D=x(y_1-y_2)-y(x_1-x_2)+(x_1y_2-x_2y_1)=xy_1-xy_2-yx_1+yx_2+x_1y_2-x_2y_1=0\\y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)=0$
This is the equation we expected.