Which one of the following is an ordered pair in R

Question

Let R be a relation on Z( the set of integers) defined by

(x,y)ER iff x^2 - y^2 = 3K for some kE Z(the set of integers)

Answer 1

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).