Question

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

Answer 1

Distace between the points (x, y) and (a+b, b-a) & (a-b, a+b) is equal
⇒ √{[x - (a + b)]² + [y - (b -a)]²} = √{x - (a - b)]² + [y - (a + b)]²}
Squaring both sides:
⇒ x² + (a + b)² - 2x(a + b) + y² + (b - a)² - 2y(b - a) = x² + (a - b)² - 2x(a - b) + y² + (a + b)² - 2y(a + b)
⇒ -2ax - 2bx - 2by + 2ay = - 2ax + 2bx - 2ay - 2by
⇒ ay - bx = bx - ay
⇒ 2ay = 2bx
⇒ bx = ay