College Algebra, Chapter 8, 8.4, Section 8.4, Problem 20

Find an equation for the conic whose graph is shown.


The ellipse $\displaystyle \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ has center on $(h, k)$ and has vertical transverse, the length of its major axis is $2a$ while the length of its minor axis is $2b$. Based from the graph the ellipse has center on $(2, -3)$ and the length if the major axis and minor axis is $6$ and $4$ respectively. Thus, $2a = 6$ and $2b = 4$, this gives us $a = 3$ and $b = 2$. Therefore, the equation is $\displaystyle \frac{(x - 2)^2}{2^2} + \frac{(y - (-3)^2)}{3^2} = 1$

$\displaystyle \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{9} = 1$