One possible condition for the three points (a,b), (b,a) and a2,−b2 to be collinear is

Given points will be collinear, ifab1ba1a2−b21=0⇒ab1b−aa−b0a2−a−b2−b0=0 Applying R2→R2−R1,R3→R3−R1⇒ (a−b)ab1−110a2−a −b2−b0=0⇒ (a−b)b2+b−a2+a=0⇒ (a−b)(a+b)−a2−b2=0⇒ (a−b)(a+b)(1−a+b)=0⇒ a=b or, a+b=0 or a=1+b