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Question
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.
Solution
The co-ordinates of a point which divided two points `(x_1,y_1)` and `(x_2,y_2)` internally in the ratio m:n is 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`
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