6) For which real values of $\alpha$ does there exist a unique pair of real numbers $\left\{x^{2}+y^{2}=\alpha\right.$ such that $(x, y)$ is a solution to the system below?
(A) $\alpha=0$
(B) $\alpha=3$
(C) $\alpha>0$
(D) $\alpha=\sqrt{3}$
(E) $\alpha \neq 0$.
$x=3 y$
6) The answer is (A). If $\alpha=0$ the first equation has the unique solution $(0,0)$, which also solves the second. If $\alpha>0$ the solutions are two, as can be seen with a geometric interpretation: they are the intersections of the circle centered at $(0,0)$ with radius $\sqrt{\alpha}$ and a line passing through the center.