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Question
If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
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Solution
The distance d between two points `(x_1,y_1)` ``nd `(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.
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