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.

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)))`

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.