Let the circle x 2+y 2 = k.

Then x 2+y 2 ≥ k represent that peripheral region of the circle x 2+y 2 = k.

According to the topic, all points on the hyperbola are in the peripheral area of the circle or tangent to the circle.

Make a straight line y=x, and the hyperbola intersects (1, 1) and (-1,-1).

It is easy to know that when a circle is tangent to a hyperbola, its radius is √2.

That is kmax=2.

Another method:

Because xy= 1

Y= 1/x

X 4-kx 2+1≥ 0 when substituting inequality.

Let x 2 = t

There is t 2-kt+ 1 ≥ 0.

Obviously, when ⊿≤0 is put forward, the above inequality holds.

That is, k 2-4 ≤ 0.

That is, -2≤k≤2.

That is kmax=2.