In what Ratio the line joining (– 1, 1) and (5, 7) is divided by the

1.	In what Ratio the line joining (– 1, 1) and (5, 7) is divided by the line x + y = 4?
Answer» Let the points be A(x₁, y₁) = (– 1, 1) and B(x₂, y₂) = (5, 7) and P(x₃, y₃) be the point which divides AB in the ratio m: n. ∴ Co-ordinates of P are \(\big(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\big)\) = \(\big(\frac{5m - n}{m + n}, \frac{7m + n}{m + n}\big)\) Since the point P lies on line x + y = 4. ∴ \(\frac{5m - n}{m + n}, \frac{7m + n}{m + n}\) = 4 ⇒ \(\frac{12m}{m+n}\) = 4 ⇒ 8m = 4n ⇒ m: n = 1: 2.

In what Ratio the line joining (– 1, 1) and (5, 7) is divided by the line x + y = 4?

Answer»

Let the points be A(x₁, y₁) = (– 1, 1) and B(x₂, y₂) = (5, 7) and P(x₃, y₃) be the point which divides AB in the ratio m: n.

∴ Co-ordinates of P are \(\big(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\big)\)

= \(\big(\frac{5m - n}{m + n}, \frac{7m + n}{m + n}\big)\)

Since the point P lies on line x + y = 4.

∴ \(\frac{5m - n}{m + n}, \frac{7m + n}{m + n}\) = 4

⇒ \(\frac{12m}{m+n}\) = 4

⇒ 8m = 4n

⇒ m: n = 1: 2.