Find the relation between x and y such that the point P(x, y) is equidistant for points A(7, 1) and B(3, 5) .

Let P(x, y) be equidistant from the points A(7, 1) and B(3, 5).

Given, AP = BP

Proof:

By distance formula,

Squaring both the sides, we get

(x - 7)² + (y - 1)² = (x - 3)² + (y - 5)²

⇒ x² + 49 - 14x + y² + 1 - 2y = x² + 9 - 6x + y² + 25 - 10y

⇒ -14x + 50 – 2y = -6x + 34 – 10y

Divide by 2, we get

⇒ -7x + 25 - y = -3x + 17 - 5y

⇒ -7x + 3x + 25 – 17 – y + 5y = 0

⇒ -4x + 8 + 4y = 0

Divide by 4, we get

⇒ x - y = 2