Prove that the equation of the line through the distinct poi

Prove that the equation of the line through the distinct points (a_1, b_1) and (a_2, b_2) can also be written as |x a_1 a_2 y b_1 b_2 1 1 1| = 0.

Solving the given equation as:

x (b1*1 - b2*1) - y (a1*1 - a2*1) + 1 (a1*b2 - a2b1) = 0

x (b1 - b2) -y (a1 - a2 ) + a1b2 - a2b1 = 0 ------------ (1)

Now let us find the equation of the line passing through points (a1, b1) and (a2, b2).

I am using the point slope equation as:

y - b1 = m (x - a1)

y - b1 = (b1-b2)/(a1-a2) * (x - a1)

Multiply both the sides by (a1 - b1), we get:

(y-b1)(a1-a2) = (x-a1)(b1-b2)

y (a1 - a2) - b1a1 + a2b1 = x (b1 - b2) - a1b1 + a1b2

x (b1 - b2) - y (a1 - a2) + a1b2 - a2b1 = 0 -------------- (2)

Comparing equation (1) and (2), we can see that both are equal.

Hence Proved.