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.