A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.

We know that the coordinates of a point dividing the line segment joining the points (x₁, y₁) and (x₂, y₂) internally in the ratio m: n are .

∴

We know that slope, m =

∴ m =

We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.

∴ m = (-1/m) =

We know that the point (x, y) lies on the line with slope m through the fixed point (x₀, y₀), if and only if, its coordinates satisfy the equation y – y₀ = m (x – x₀)

Here, the point is.

∴

⇒ 3((1 + n) y – 3) = (-(1 + n) x + 2 + n)

⇒ 3(1 + n) y – 9 = - (1 + n) x + 2 + n

⇒ (1 + n) x + 3(1 + n) y – n – 9 – 2 = 0

⇒ (1 + n) x + 3(1 + n) y – n – 11 = 0

Ans. The equation of the line is (1 + n) x + 3(1 + n) y – n – 11 = 0.