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

Answer:

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

It is given that PQ = PR  PQ2 = PR2

{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 – 2y(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