Question

If the point x y is equidistant from the point a minus b a + b and minus a minus b a + b prove that x minus a is equal to zero

Answer 1

P* A=P* B

Taking square on both sides,

PA²=PB²

now by using distance formula,,  

{x-(a+b)}^2+{y-(b-a)}^2={x-(a-b)}^2+{y-(a+b)}^2

-> x^2+ (a+b)^2 -2x* (a+b) +y^2 +(b-a)^2 -2y*(b-a)* y= x^2+(a-b)^2-2x(a-b)+y^2+(a+b)^2-2y(a+b)

->2x* (a-b) -2x* (a+b)= 2y* (b-a) -2y* (a+b)

->2x* [a -b -a -b]=2y*[ b-a-a-b]

-> 2x* (-2b) = 2y* (-2a)

-> b* x = a* y