Question

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

Answer 1

As per the question, we have

`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`      (Squaring both sides) 

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

`⇒-x(a+b) - y (a-b) = -x(a-b) -y(a+b)`

`⇒-xa -xb -ay +by = -xa + bx -ya-by`

⇒ by=bx

Hence, . bx = ay