Question

In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

Answer 1

The ratio in which the y-axis divides two points `(x_1,y_1)` and (x_2,y_2) is λ : 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(3,7).

Since the point is on the y-axis so, x coordinate is 0.

`(3lambda - 2)/1 = 0`

`=> lambda = 2/3`

Thus the given points are divided by the y-axis in the ratio 2:3

The coordinates of this point (x, y) can be found by using the earlier mentioned formula.

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

`(x,y) = ((((6 - 2(3))/3)/((2+3)/3)) "," (((14 - 3(3))/3)/((2+3)/3)))`

`(x,y) = ((0/5)","(5/5))`

(x,y) = (0,1)

Thus the co-ordinates of the point which divides the given points in the required ratio are (0,1)