Solving Equations – Vectorize NSolve

F[x_, y_, a_] : = (a + x) ^ 2 - y ^ 2
Fpartx[x_, a_] = D[F[x, y, a], x]Fparty[y_, a_] = D[F[x, y, a]y]

For toy list a,

a = range[10] (* a = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} *)

table[NSolve[
Fpartx[x, a[[i]]]== 0 && Fparty[y, a[[i]]]== 0, {x, y}]{1, 1,
10}]

{{{x -> -1., y -> 0.}}, {{x -> -2., y -> 0.}}, {{x -> -3.,
y -> 0.}}, {{x -> -4., y -> 0.}}, {{x -> -5.,
y -> 0.}}, {{x -> -6., y -> 0.}}, {{x -> -7.,
y -> 0.}}, {{x -> -8., y -> 0.}}, {{x -> -9.,
y -> 0.}}, {{x -> -10., y -> 0.}}}

Do what you wanted?