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प्रश्न
Determine the ratio in which the point P (m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.
Determine the ratio in which the point P(m, 6) divides the line segment joining the points A (−4, 3) and B (2, 8). Also, find the value of m.
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उत्तर
The co-ordinates of a point which divides two points (x1, y1) and (x2, y2) internally in the ratio m : n are given by the formula,
`(x, y) = ((mx_2 + nx_1) / (m + 2))"," ((my_2 + ny_1) / (m + n))`
Here, we are given that the point P(m, 6) divides the line joining the points A(−4, 3) and B(2, 8) in some ratio.
Let us substitute these values in the earlier-mentioned formula.
`(m, 6) = ((m(2) + n(-4)) / (m + n)), ((m(8) + n(3)) / (m + n))`
Equating the individual components, we have
`6 = ((m(8) + n(3)) / (m + n))`
6m + 6n = 8m + 3n
2m = 3n
`m / n = 3 / 2`
We see that the ratio in which the given point divides the line segment is 3 : 2.
Let us now use this ratio to find out the value of m.
`(m, 6) = ((m(2) + n(4)) / (m = n)), ((m(8) + n(3)) / (m + n))`
`(m, 6) = ((3(2) + 2(-4)) / (3 + 2)), ((3(8) + 2(3)) / (3 + 2))`
Equating the individual components, we have
`m = (3(2) + 2(4)) / (3 + 2)`
`m = -2 / 5`
Thus, the value of m is `- 2 / 5` and m : n = 3 : 2.
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