Question

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

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 x-axis divides these points is `(6lambda - 3)/3 = 0`

`lambda = 1/2`

Let point P(x, y) divide the line joining ‘AB’ in the ratio 1 : 2

Substituting these values in the earlier mentioned formula, we have

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

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

`(x,y) = ((1/3)","(0/3))`

`(x,y) = (1/3, 0)`

Thus the ratio in which the x−axis divides the two given points and the co-ordinates of the point is `(1/3, 0)`.