College Algebra, Chapter 8, 8.4, Section 8.4, Problem 24

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.


$\begin{equation} \begin{aligned} 9x^2- 36x + 4y^2 =& 0 && \text{Factor and group terms} \\ \\ 9(x^2 - 4x + ) + 4y^2 =& 0 && \text{Complete the square: add } \left( \frac{4}{2} \right)^2 = 4 \text{ on the right side and 36 on the left side} \\ \\ 9(x^2 - 4x + 4) + 4y^2 =& 36 && \text{Complete the square} \\ \\ 9(x - 2)^2 + 4y^2 =& 36 && \text{Divide by } 36 \\ \\ \frac{(x - 2)^2}{4} + \frac{y^2}{9} =& 1 && \end{aligned} \end{equation}$


We can say that the equation is an ellipse since it is the sum of the squares. The equation $\displaystyle \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ has a vertical major axis and vertices on $(0, \pm a)$, center on $(h, k)$ and foci $(0, \pm c)$ where $c = \sqrt{a^2 - b^2}$.

Also, the length of its major and minor axis is $2a$ and $2b$ respectively, So if $a^2 = 9$ and $b^2 = 4$, then $a = 3, b = 2$ and $c = \sqrt{9 - 4} = \sqrt{5}$.

Thus, by applying transformations,

center $(2,0)$

vertices $(2, 0 \pm 3) \to (2, \pm 3)$

foci $(2, 0 \pm \sqrt{5}) \to (2, \pm \sqrt{5})$

length of major axis $6$

length of minor axis $4$

Therefore, the graphs is