Given points are A(a + b, b - a) and B(a - b, a + b). It is told that S(x, y) is equidistant from A and B.
So, we get SA = SB,
We know that distance between two points (x1, y1) and (x2, y2) is
Now,
⇒ SA = SB
⇒ SA2 = SB2
⇒ (x - (a + b))2 + (y - (b - a))2 = (x - (a - b))2 + (y - (a + b))2
⇒ x2 - 2(a + b)x + (a + b)2 + y2 - 2(b - a)y + (b - a)2 = x2 - 2(a - b)x + (a - b)2 + y2 - 2(a + b)y + (a + b)2
⇒ x(- 2a - 2b + 2a - 2b) = y(2b - 2a - 2a - 2b)
⇒ x(- 4b) = y(- 4a)
⇒ x(b) = y(a)
⇒ x/y = a/b
Applying componendo and dividendo,
∴ Thus proved.