19. Consider the curves $y=A x^{2}$ and $y^{2}+3=x^{2}+4 y$, where $A$ is a positive constant, and $x$ and $y$ are real variables. How many intersection points do these two curves have?
(A) Exactly 4;
(B) Exactly 2;
(C) At least 1, and the number of intersection points varies with different values of $A$;
(D) For at least one positive value of $A$, the number of intersection points is 0;
(E) None of the above conclusions are correct.
19. The second equation can be rewritten as $(y-2)^{2}-x^{2}$ $=1$, which represents a hyperbola centered at $(0,2)$, with vertices at $(0,1)$ and $(0,3)$, and asymptotes given by $y=2 \pm x$. When $A>0$, the graph of $y=A x^{2}$ is a parabola opening upwards with its vertex at the origin. Therefore, the parabola intersects the lower branch of the hyperbola at two distinct points. The upper branch of the hyperbola is given by $y=2+\sqrt{x^{2}+1}$. For any given positive number $A$, if $|x|$ is sufficiently large, then $A x^{2}>2+\sqrt{x^{2}+1}$ $\approx 2+x$. On the other hand, when $x=0$, $A x^{2}<2+\sqrt{x^{2}+1}$. Thus, for any given positive number $A$, the parabola must intersect the upper branch of the hyperbola at two distinct

points.
(A) True.