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Question
Let A(2, 3) and B(2, −4) be two points. If P lies on the x-axis, such that AP = `3/7` AB, find the coordinates of P.
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Solution
Given points are A(2, 3) and B(2, −4)
The point P lies on the x-axis.
∴ The point P is (x, 0)
AP = `3/7"AB"`
`"AP"/"AB" = 3/7`
`"AP"/"PB" = 3/4` ...(1)
Distance = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
AP = `sqrt((x - 2)^2 + (0 - 3)^2`
= `sqrt(x^2 - 4x + 4 + 9)`
= `sqrt(x^2 - 4x + 13)`
BP = `sqrt((x - 2)^2 + (0 + 4)^2`
= `sqrt(x^2 - 4x + 4 + 16)`
= `sqrt(x^2 - 4x + 20)`
From (1) we get
`"AP"/"PB" = 3/4`
`(sqrt(x^2 - 4x + 13))/(sqrt(x^2 - 4x + 20)) = 3/4` ...(squaring ob both sides)
`(x^2 - 4x + 13)/(x^2 - 4x + 20) = 9/16`
16x2 – 64x + 208 = 9x2 – 36x + 180
16x2 – 9x2 – 64x + 36x + 208 – 180 = 0
7x2 – 28x + 28 = 0
x2 – 4x + 4 = 0
(x – 2)2 = 0
x – 2 = 0
x = 2
∴ The point P is (2, 0)
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