# How do you graph the ellipse (x-4)^2/8+(y-2)^2/18=1 and find the center, the major and minor axis, vertices, foci and eccentricity?

May 2, 2018

Center: $4 , 2$
Major axis: length $3 \sqrt{2}$, vertical
Minor axis: legnth $2 \sqrt{2}$, horizontal
Vertices: $\left(2 \sqrt{2} , 0\right) , \left(- 2 \sqrt{2} , 0\right) , \left(0 , 3 \sqrt{2}\right) , \left(0 , - 3 \sqrt{2}\right)$
Foci: $\left(0 , - \sqrt{10}\right) , \left(0 , \sqrt{10}\right)$
Eccentricity: $\frac{\sqrt{5}}{3}$

#### Explanation:

Given the general formula

${\left(x - {x}_{0}\right)}^{2} / \left({a}^{2}\right) + {\left(y - {y}_{0}\right)}^{2} / \left({b}^{2}\right) = 1$

you have that:
- ${x}_{0}$ is the $x$ component of the center
- ${y}_{0}$ is the $y$ component of the center
- $a$ is the horizontal semi-axis
- $b$ is the vertical semi-axis

• Foci: $c = \sqrt{{b}^{2} - {a}^{2}}$ (in general, major axis minus minor axis)
• Eccentricity = $\frac{c}{b}$ (in general, focus divided by major axis)