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Question
A line AB meets X-axis at A and Y-axis at B. P(4, –1) divides AB in the ration 1 : 2.
- Find the co-ordinates of A and B.
- Find the equation of the line through P and perpendicular to AB.
Solution
i. Since A lies on the X-axis, let the co-ordinates of A be (x, 0).
Since B lies on the Y-axis, let the co-ordinates of B be (0, y).
Let m = 1 and n = 2
Using section formula,
Coordinates of P = `((1(0) + 2(x))/(1 + 2), (1y + 2(0))/(1 + 2))`
`=> (4, -1) = ((2x)/3, y/3)`
`=> (2x)/3 = 4` and `y/3 = -1`
`=>` x = 6 and y = –3
So, the co-ordinates of A are (6, 0) and that of B are (0, –3).
ii. Slope of AB = `(-3 - 0)/(0 - 6) = (-3)/(-6) = 1/2`
`=>` Slope of line perpendicular to AB = m = –2
P = (4, –1)
Thus, the required equation is
y – y1 = m(x – x1)
`=>` y – (–1) = –2(x – 4)
`=>` y + 1 = –2x + 8
`=>` 2x + y = 7
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