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Consider the equation 
\[ ax^2 + bxy + cy^2  = 1 \] for three parameters $a$, $b$, and $c$.
\begin{enumerate}
    \item This is the equation corresponding to graphs of conic sections. What kind of a shape is drawn if:
    \begin{enumerate}
        \item $b^2 - 4ac < 0$?
        \item $b^2 - 4ac = 0$?
        \item $b^2 - 4ac > 0$?
    \end{enumerate}
    You can use a calculator here to draw graphs and identify the curves from that.
    \item Find $\frac{dy}{dx}$ for the first equation in this problem. Your answer will depend on the parameters $a$, $b$, and $c$. (You should solve for $\frac{dy}{dx}$ here. It'll make your life a lot easier.)
    \item Set $\frac{dy}{dx} = 0$ and see what this means in terms of the values of $x$ and $y$. Plug this into the initial equation to see what that means in terms of where horizontal tangent lines exist.
    \item Is it possible to choose values for $a$, $b$, and $c$ so that the graph both is a hyperbola \textit{and} there is a point where the graph has a horizontal tangent line? If so, come up with values for $a$, $b$, and $c$ that do this, as well as the point $(x,y)$ that has the horizontal tangent line. If not, explain why not.
    \item Is it possible to choose values for $a$, $b$, and $c$ so that the graph both is a hyperbola \textit{and} there are no points where the graph has a horizontal tangent line? If so, come up with values for $a$, $b$, and $c$ that do this. If not, explain why not.
    \item Is it possible to choose values for $a$, $b$, and $c$ so that the graph both is an ellipse \textit{and} there is a point where the graph has a horizontal tangent line? If so, come up with values for $a$, $b$, and $c$ that do this, as well as the point $(x,y)$ that has the horizontal tangent line. If not, explain why not.
    \item Is it possible to choose values for $a$, $b$, and $c$ so that the graph both is an ellipse \textit{and} there are no points where the graph has a horizontal tangent line? If so, come up with values for $a$, $b$, and $c$ that do this. If not, explain why not.
\end{enumerate}

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