The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach.
Suppose $b^2-4ac>0$.
$(1)$ Show that one of the following occurs: $\{y\ |∆_x(y)\ge 0\} =\mathbb R$ and $∆_x(y)\ne 0,\ \{y\ |∆_x(y)=0\}=\{y_o\}$ and $\{y\ |∆_x(y)>0\}=\{y\ |\ |y|>y_o\}$, or there exist real numbers $\alpha$ and $\beta$, $\alpha<\beta$, such that $\{y\ |∆_x(y)\ge 0\}= \{y\ |\ y\le\alpha\}\cup\{y\ |\ y\ge\beta\}$.
$(2)$ Show that if there exist real numbers $\alpha$ and $\beta$, $\alpha<\beta$, such that $\{y\ |∆_x(y)\ge 0\}=\{y\ |\ y\le \alpha\}\cup\{y\ |\ y\ge \beta\}$, then $V(P)$ is a hyperbola.
I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ has the discriminant $\Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$. And I stuck!
Any hint to start solving this?