Question

Use the fact that :

0= x y 1

a1 b1 1

a2 b2 1

to determine the equation of the line passing through the distinct points (a1, b1) and (a2, b2), where |·| stands for det(·), the determinant. 

Answer 1

$$\begin{vmatrix} x & y & 1 \\ a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \end{vmatrix}=0$$

$$x\begin{vmatrix} b_1& 1 \\ b_2 & 1 \end{vmatrix}-y\begin{vmatrix} a_1 & 1 \\ a_2 & 1 \end{vmatrix}+1\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}=0$$

$$x(b_1-b_2)-y(a_1-a_2)+a_1b_2-a_2b_1=0$$

The equation of the line passing through the distinct points $(a_1, b_1)$ and $(a_2, b_2)$ in slope-intercept form

$$y=\dfrac{b_1-b_2}{a_1-a_2}x+\dfrac{a_1b_2-a_2b_1}{a_1-a_2}, a_1\not=a_2$$