Question

If the intercept of a line between the coordinate axes is divided by the point (–5, 4) in the ratio 1 : 2, then find the equation of the line.

Answer 1

Let a and b be the intercepts on the given line.

∴ Coordinates of A and B are (a, 0) and (0, b) respectively

∴ – 5 = `(1 xx 0 + 2 xx a)/(1 + 2)`

⇒ 2a = – 15      .......`[(because "X" = (m_1x_2 + m_2x_1)/(m_1 + m_2)),("and"  "Y" = (m_1y_2 +  m_2y_1)/(m_1 +  m_2))]`

⇒ a = ` (-15)/2`

∴ A = `((-15)/2, 0)`

And  4 = `(1 xx b + 0 xx 2)/(1 + 2)`

⇒ 4 = `b/3`

⇒ b = 12

∴ B = (0, 12)

So, the equation of line AB is

y – y₁ = `(y_2 - y_1)/(x_2 - x_1) (x - x_1)`

y – 0 = `((12 - 0)/(0 + 15/2)) (x + 15/2)`

⇒ y = `(12 xx 2)/15 (x + 15/2)`

⇒ y = `8/5(x + 15/2)`

⇒ 5y = 8x + 60

⇒ 8x – 5y + 60 = 0

Hence, the required equation is 8x – 5y + 60 = 0.