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.

Answer 1

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)