#### Question

Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.

#### 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)))`

Here we are given that the point *P*(2*,y*) divides the line joining the points *A*(−2*,*2) and *B*(3*,*7) in some ratio.

Let us substitute these values in the earlier mentioned formula.

`(2,y) = (((m(3) +n(-2))/(m + n))"," ((m(7)+n(2))/(m+n)))`

Equating the individual components we have

`2 = (m(3) + n(-2))/(m + n)`

2m + 2n = 3m - 2n

m - 4n

`m/n = 4/1`

We see that the ratio in which the given point divides the line segment is 4: 1.

Let us now use this ratio to find out the value of ‘*y*’.

`(2,y) = (((m(3) + n(-2))/(m + n))"," ((m(7) + n(2))/(m + n)))`

`(2,y) = (((4(3) + 1(-2))/(4 +1))","((4(7) + 1(2))/(4 +1)))`

Equating the individual components we have

`y = (4(7) + 1()2)/(4 + 1)`

y = 6

Thus the value of ‘*y*’ is 6