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Question
Determine the ratio in which the straight line x - y - 2 = 0 divides the line segment
joining (3, -1) and (8, 9).
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Solution
Let the line x - y - 2 = 0 divide the line segment joining the points A (3,−1) and B (8, 9) in the ratio λ:1 at any point P(x,y)
Now according to the section formula if point a point P divides a line segment joining `A (x_1, y_1)` and `B(x_2,y_2)` the ratio m: n internally than.
`P(x,y)= ((nx_2 + mx_2)/(m + n), (ny_1 + my_2)/(m + n))`
So,
`P(x,y) = ((8y + 3)/(lambda + 1), (9lambda - 1)/(lambda+ 1))`
Since, P lies on the given line. So,
x - y - 2 =0
Put the values of co-ordinates of point P in the equation of line to get,
`((8lambda + 3)/(lambda + 1)) - ((9lambda- 1)/(lamda + 1)) - 2 = 0`
On further simplification we get,
`-3lambda + 2 = 0`
So, `lambda =2/3`
So the line divides the line segment joining A and B in the ratio 2: 3 internally.
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