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.

Answer 1

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`