If a point P(x, y) is equidistant from the points A(6, -1) and B(2, 3)

1.	If a point P(x, y) is equidistant from the points A(6, -1) and B(2, 3), find the relation between x and y.
Answer» Given: Point P(x, y) is equidistant from points A(6, -1) and B(2, 3) i.e., distance of P from A = distance of P from B ⇒ \(\sqrt{(x-6)^2+(y+1)^2}\) = \(\sqrt{(x-2)^2+(y-3)^2}\) Squaring both sides, ⇒ (x – 6)² + (y – 1)² = (x – 2)² + (y – 3)² ⇒ x² – 12x + 36 + y² – 2y + 1 = x² – 4x + 4 + y² – 6y + 9 ⇒ –12x + 36 + 2y + 1 = – 4x + 4 – 6y + 9 ⇒ – 8x + 8y = –24 ⇒ x – y = 3 Therefore, x – y = 3 is the required relation.

If a point P(x, y) is equidistant from the points A(6, -1) and B(2, 3), find the relation between x and y.

Answer»

Given: Point P(x, y) is equidistant from points A(6, -1) and B(2, 3)

i.e., distance of P from A = distance of P from B

⇒ \(\sqrt{(x-6)^2+(y+1)^2}\) = \(\sqrt{(x-2)^2+(y-3)^2}\)

Squaring both sides,

⇒ (x – 6)² + (y – 1)² = (x – 2)² + (y – 3)²

⇒ x² – 12x + 36 + y² – 2y + 1 = x² – 4x + 4 + y² – 6y + 9

⇒ –12x + 36 + 2y + 1 = – 4x + 4 – 6y + 9

⇒ – 8x + 8y = –24

⇒ x – y = 3

Therefore, x – y = 3 is the required relation.