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