Question

P(x, y) is a point equidistant from the points A(4, 3) and B(3, 4). Prove that x − y = 0.

Answer 1

Given, P(x, y), A(4, 3) and B(3, 4)

PA = PB  ...(Equidistant)

So, by distance formula, we have

Distance between two points = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

PA² = PB²

⇒ (4 − x)² + (3 − y)² = (3 − x)² + (4 − y)²

⇒ (4)² + (x)² − 2 × 4 × x + (3)² + (y)² − 2 × 3 × y = (3)² + (x)² − 2 × 3 × x + (4)² + (y)² − 2 × 4 × y

⇒ 16 + x² − 8x + 9 + y² − 6y = 9 + x² − 6x + 16 + y² − 8y

⇒ −8x + 6x − 6y + 8y = 0

⇒ −2x + 2y = 0

⇒ −2(x − y) = 0

⇒ x − y = 0