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.

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 + 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