Question

In what ratio does the point `(24/11, y)` divide the line segment joining the points P(2, –2) and Q(3, 7)? Also find the value of y.

Answer 1

Let the point P`(24/11, y)` divide the line PQ in the ratio k : 1.

Then, by the section formula:

`x = (mx_2+nx_1)/(m+n), y = (my_2 + ny_1)/(m + n)`

The coordinates of R are `(24/11, y)`

`24/11 = (3k + 2)/(k + 1), y = (7k - 2)/(k + 1)`

`=>24(k + 1) = 33k + 22, y(k + 1)= 7k - 2`

⇒24k + 24 = 33k + 22  ,   yk + y =7k − 2

⇒2 = 9k

`=> k = 2/9`

Now consider the equation yk + y = 7k - 2 and put `k = 2/9`

`=> 2/9y + y = 14/9 - 2`

`=> 11/9y = (-4)/9`

`=> y = (-4)/11`

Therefore, the point R divides the line PQ in the ratio 2 : 9

And, the coordinates of R are `(24/11, (-4)/11)`