Question

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

Answer 1

The equation of the line joining the points (–1, 1) and (5, 7) is given by

y - 1 = `(7 -1)/(5 + 1) (x + 1)`

y - 1 = `6/6 (x + 1)`

x - y + 2 = 0    ....(1)

The equation of the given line is

x + y – 4 = 0   …(2)

The point of intersection of lines (1) and (2) is given by

x = 1 and y = 3

Let point (1, 3) divide the line segment joining (–1, 1) and (5, 7) in the ratio 1:k. Accordingly, by section formula,

(1, 3) = `((k(-1) + 1(5))/(1 + k), (k (1) + 1 (7))/(1 + k))`

= (1, 3) = `((-k + 5)/(1 + k), (k + 7)/(1 + k))`

= `(-k + 5)/(1 + k) = 1, (k + 7)/(1 + k) = 3`

∴ `(-k + 5)/(1 + k) = 1`

= -k + 5 = 1 + k

= 2k = 4

= k = 2

Thus, the line joining the points (–1, 1) and (5, 7) is divided by line

x + y = 4 in the ratio 1:2.