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.