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Question
Point R (h, k) divides a line segment between the axes in the ratio 1:2. Find equation of the line.
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Solution
Let AB be the line segment between the axes such that point R (h, k) divides AB in the ratio 1: 2.
Let the respective coordinates of A and B be (x, 0) and (0, y).
Since point R (h, k) divides AB in the ratio 1: 2, according to the section formula,
(h, k) = `(1 xx 0 + 2 xx x)/(1 + 2), (1 xx y + 2 xx 0)/(1 + 2)`
= (h, k) = `((2x)/3, y/3)`
= `h = (2x)/3 and k = y/3`
= x = `(3h)/2 and y = 3k`
Therefore, the respective coordinates of A and B are `((3h)/2,0)` and (0, 3k).
Now, the equation of line AB passing through points `((3h)/2,0)` and (0, 3k) is
(y - 0) = `(3k - 0)/(0 - (3h)/2) (x - (3h)/2)`
y = `(2k)/h (x - (3h)/2)`
hy = `-(2k)/h (x - (3h)/2)`
hy = -2kx + 3hk
i.e., 2kx + hy = 3hk
Thus, the required equation of the line is 2kx + hy = 3hk.
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