Question

Find the equation of the line joining (3, 1) and (9, 3) using the determinants.

Answer 1

Let A(3, 1) and B(9, 3) be two fixed points.

Let P(x, y) be any point lying on the line joining A and B.

If A, P and B are collinear, the area of ΔAPB must be zero.

We use the determinant formula for the area of a triangle formed by three points:

Area of triangle = `1/2 |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|`

x₁ = 3, y₁ = 1, x₂ = 9, y₂ = 3, x₃ = x, y₃ = y

⇒ `1/2 |(3,1,1),(9,3,1),(x,y,1)| = 0`

⇒ `1/2 [3|(3,1),(y, 1)| - 1|(9,1),(x,1)| + 1|(9,3),(x,y)|] = 0`

⇒ `1/2[3(3 - y) - 1(9 - x) + 1(9y - 3x)] = 0`

⇒ `1/2 [(9 - 3y) + (-9 + x) + (9y - 3x)] = 0`

⇒ `1/2 [0 + x - 3y + 9y - 3x] = 0`

⇒ `1/2 [-2x + 6y] = 0`

Multiply both sides by 2 to eliminate the fraction:

−2x + 6y = 0

Simplify the equation by dividing the entire equation by 2:

−2x + 6y = 0

⇒ x − 3y = 0

Hence, x − 3y = 0 is the required line.