x2-y2=3K, y2=x2-3K, so all ordered pairs are (x,±√(x2-3K)). This requires x2≥3K because x,y∈ℝ. That is, complex numbers are excluded.
x2≥3K⇒x≥√(3K) or x≤-√(3K). This dependence on K shifts the domain, opening up a no-go zone centred at the origin (when graphed).
If K=3, x≥3 or x≤-3. An example is x=3 making ordered pairs (3,0) and (-3,0). Another set of ordered pairs is (5,4), (-5,4), (5,-4) and (-5,-4). Each value of K≠0 always produces 4 related ordered pairs, provided x2≥3K.
The special case is when K=0, then y=x and y=-x are two perpendicular lines passing through the origin. For all other values of K the result is a hyperbola (vertical when K<0 and horizontal when K>0).