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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 in each case.
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 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)`
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