Question

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by y-axis. Also, find the coordinates of the point of division in each case.

Answer 1

The ratio in which the x−axis divides two points `(x_1,y_1)` and `(x_2,y_2)` is λ : 1

The ratio in which the y-axis divides two points `(x_1,y_1)` and `(x_2,y_2)` is μ : 1

The coordinates of the point dividing two points `(x_1,y_1)`  and `(x_2,y_2)` in the ratio m:n is given as,

`(x,y) = (((lambdax_2 + x_1)/(lambda + 1))","((lambday_2 + y_1)/(lambda + 1)))` Where `lambda = m/n`

Here the two given points are A(−2,−3) and B(5,6).

The ratio in which the y-axis divides these points is `(5mu - 2)/3 = 0`

`=> mu= 2/5`

Let point P(x, y) divide the line joining ‘AB’ in the ratio 2: 5

Substituting these values in the earlier mentioned formula we have,

`(x,y) = (((2/5(5) + (-2))/(2/5 + 1))","((2/5(6) + (-3))/(2/5 + 1)))`

`(x,y) = ((((10 + 5(-2))/5)/((2 + 5)/5)) "," (((12 + 5(-3))/3)/((2 + 5)/5)))`

`(x,y) = ((0/7)","(- 3/7))`

`(x,y) = (0, - 3/7)`

Thus the ratio in which the x-axis divides the two given points and the co-ordinates of the point is 2:5 and `(0, - 3/7)`