Question

If the point P(x, y) is equidistant from the points A (5, 1) and B (1, 5), prove that x = y.

Point P (x, y) is equidistant from points A (5, 1) and B (1, 5). Prove that x = y.

Answer 1

The distance d between two points `(x_1,y_1)` and `(x_2,y_2)` is given by the formula

`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`

The three given points are P(x, y), A(5, 1) and B(1, 5).

Now let us find the distance between ‘P’ and ‘A’.

`PA = sqrt((x - 5)^2 + (y - 1)^2)`

Now, let us find the distance between ‘P’ and ‘B’.

`PB = sqrt((x - 1)^2 + (y - 5)^2)`

It is given that both these distances are equal. So, let us equate both the above equations,

PA = PB 

`sqrt((x - 5)^2 + (y -1)^2) = sqrt((x - 1)^2 + (y - 5)^2)`

Squaring on both sides of the equation we get,

`(x - 5)^2 + (y - 1)^2 = (x - 1)^2 + (y - 5)^2`

`=> x^2 + 25 - 10x + y^2 + 1 - 2y = x^2 + 1 - 2x + y^2 + 25 - 10y` 

`=> 26 - 10x - 2y = 26 - 10y - 2x`

`=> 10y - 2y = 10x - 2x`

`=> 8y = 8x`

`=> y = x`

Hence we have proved that x = y.