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Question
Determine the ratio in which the point (-6, a) divides the join of A (-3, 1) and B (-8, 9). Also, find the value of a.
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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 + n))"," ((my_2 + ny_1)/(m + n)))`
Let us substitute these values in the earlier mentioned formula.
`(-6, a) = (((m(-8) +n(-3))/(m + n))","((m(9) +n(1))/(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 'a'.
`(-6,a) = (((m(-8) + n(-3))/(m - n))","((m(9) + n(1))/(m + n)))`
`(-6, a) = (((3(-8) + 2(-3))/(3 + 2))"," ((3(9) + 2(1))/(3 + 2)))`
Equating the individual components we have
`a = (3(9) + 2(1))/(3 + 2)`
`a= 29/5`
Thus the vlaue of a is 29/5
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