find the eqyuation of hyperbola traced by points that moves so the difference to (0,0) and (11,11) is 11

Answer 1

just plug in your specification:
√(x^2+y^2) = √((x-11)^2 + (y-11)^2) + 11
square both sides to get
x^2+y^2 = (x-11)^2 + (y-11)^2 + 121 + 22√((x-11)^2 + (y-11)^2)
collect terms to get and divide by 11 to get
2x+2y-33 = 2√((x-11)^2 + (y-11)^2)
square again and collect terms and you wind up with
44x + 44y - 8xy = 121

which is, as expected, a rotated hyperbola.
You could have started out by specifying that the foci were at ±11/√2 and the shifted and rotated it, but that would have been a bit more trouble.