Question

If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.

Answer 1

Let P(x, y), Q(a + b, b – a) and R (a – b, a + b) be the given points. Then

PQ = PR   ...(Given)

⇒ `sqrt({x - (a + b)}^2 + {y - (b - a)}^2) = sqrt({x - (a - b)}^2 + {y - (a + b)}^2`

⇒ `{x - (a + b)}^2 + {y - (b - a)}^2 = {x - (a - b)}^2 + {y - (a + b)}^2`

⇒ x² – 2x(a + b) + (a + b)² + y² – 2y(b – a) + (b – a)² = x² + (a – b)² – 2x(a – b) + y² – 2(a + b) + (a + b)²

⇒ –2x(a + b) – 2y(b – a) = –2x(a – b) – 2y(a + b)

⇒ ax + bx + by – ay = ax – bx + ay + by

⇒ 2bx = 2ay

⇒ bx = ay