If the distances of P(x, y) from points A(3, 6) and B(- 3, 4) are equal, prove that 3x + y = 5

Given points are A(3, 6) and B(- 3, 4). It is told that S(x, y) is equidistant from A and B.

So, we get SA = SB,

We know that distance between two points (x₁, y₁) and (x₂, y₂) is .

Now,

⇒ SA = SB

⇒ SA² = SB²

⇒ (x - 3)² + (y - 6)² = (x - (- 3))² + (y - 4)²

⇒ (x - 3)² + (y - 6)² = (x + 3)² + (y - 4)²

⇒ x² - 6x + 9 + y² - 12y + 36 = x² + 6x + 9 + y² - 8y + 16

⇒ 12x + 4y = 20

⇒ 3x + y = 5

∴ Thus proved.