Why can’t we solve this identity for the values of a, b, t?
Reduce[ForAll[{x, y}, {x > 0, y > 0, t > 0,
t^2 + 2 t x + x^2 + 2 t y + 12 + y^2 ==
a^2 + b^2 - 2 a x + x^2 - 2 b y + y^2}], {a, b, t}, Reals]
Reduce[ForAll[{x, y}, {x > 0, y > 0, x y == 6,
t > 0, (x + y + t)^2 == (x - a)^2 + (y - b)^2}], {a, b, t}, Reals]
The essence of the calculation is to equate the corresponding coefficients of each term—along with the constant term—and thereby solve for a, b, t.
Reduce[{t^2 + 12 == a^2 + b^2, 2 t == -2 a, 2 t == -2 b,
t > 0}, Reals]
