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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.
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Solution
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)`
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