College Algebra, Chapter 8, 8.4, Section 8.4, Problem 34

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices and asymptotes. Sketch the graph of the equation. If the equation has no graph, explain why.


$
\begin{equation}
\begin{aligned}

x^2 + 4y^2 + 20x - 40y + 300 =& 0
&& \text{Subtract } 300
\\
\\
x^2 + 4y^2 + 20x - 40y =& -300
&& \text{Factor and group terms}
\\
\\
(x^2 + 20x + \quad) + 4 (y^2 - 10y + \quad) =& -300
&& \text{Complete the square: add } \left( \frac{20}{2} \right)^2 = 100 \text{ or both sides and } \left( \frac{-10}{2} \right)^2 = 25 \text{ on the left and $100$ on the right}
\\
\\
(x^2 + 20x + 100) + 4(y^2 - 10y + 25) =& -300 + 100 + 100
&& \text{Perfect square}
\\
\\
(x + 10)^2 + 4(y - 5)^2 =& -100
&& \text{Divide both sides by } -100
\\
\\
\frac{-(x + 10)^2}{100} - \frac{(y - 5)^2}{100} =& 1
&&

\end{aligned}
\end{equation}
$


We can see that the equation has no solution since the sum of the squares can never be a negative value. Thus, the equation is a degenerate conic and it has no graph.