Given that k is the number of distinct ordered pair $$(x, y)$$ of real numbers satisfying equations $$x + y = xy$$ and $$x - y = 3xy$$ then k is one of the roots of the quadratic equation(s) 

This is a multiple-correct question, and the correct options are (a) and (c). Adding the equations gives $$2x = 4xy$$, so $$x = 0$$ or $$y = \frac{1}{2}$$. If $$x = 0$$ then $$y = 0$$, and if $$y = \frac{1}{2}$$ then $$x + \frac{1}{2} = \frac{x}{2}$$ gives $$x = -1$$, which also satisfies the second equation. So there are 2 ordered pairs, $$k = 2$$, and 2 is a root of $$x^2 - 3x + 2 = 0$$ (roots 1, 2) and $$x^2 - 6x + 8 = 0$$ (roots 2, 4) only.