Question

If the point px,y is equidistant from the point A(a+b,b-a) and B(a-b,b+a) then 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}

⇒ 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

Hence proved!