(1) Suppose (p,q) is a lattice point on the parabola (that is p and q are integers), then

q=ap²

and for any integer value of k it follows that

k²q = a(kp)²

so the point (kp, k²q) is another lattice point on the parabola. As k can take infinitely many values there are infinitely many lattice points on the parabola.

(2)The hyperbola has equation:

(x-y)(x+y) = 84.

Noting that (x-y) and (x+y) have the same parity (both odd or both even) and that their product is even we see that both (x-y) and (x+y) must be even. Take x-y = 2u and x+y=2v then

uv= 84
4
= 21

and so the only possibilities with u, v ł 0 are:

(u,v) = (1,21), (3,7), (7,3), (21, 1).

Now x=u+v and y=v-u giving the lattice points:

(22, 20), (10, 4), (10, -4), (22,-20)

and (by symmetry):

(-22, 20), (-10, 4), (-22, -20), (-10, -4)