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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 ratio 1 : 2.
1) Find the coordinates of A and B.
2) Find the equation of the line through P and perpendicular to AB.
Solution
1) Since A lies on the x-axis, let the coordinates of A be (x, 0).
Since B lies on the y-axis, let the coordinates 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 coordinates of A are (6,0) and that of B are (0, -3).
2) Slope of AB = `(-3-0)/(0-6) = (-3)/(-6) = 1/2`
=> Slope of line perpendicular to AB=m= -2
P = (4,-1)
⇒Required equation is
`y - y_1 = m(x - x_1)`
`=> y - (-1) = -2(x - 4)`
`=> y + 1 = -2x + 8`
`=> 2x + y = 7`
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