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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.
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Solution
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)
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