# How do you write an equation of an ellipse given the major axis is 20 units long and parallel to the y-axis and the minor axis is 6 units long, center at (4,2)?

Jan 27, 2017

The Cartesian form of the equation of an ellipse with a vertical major axis is ${\left(y - k\right)}^{2} / {a}^{2} + {\left(x - h\right)}^{2} / {b}^{2} = 1 \text{ [1]}$
where $\left(h , k\right)$ is the center, 2a is the major axis, and 2b is the minor axis.

#### Explanation:

Given that the center is $\left(4 , 2\right)$, substitute 4 for h and 2 for k into equation [1]:

${\left(y - 2\right)}^{2} / {a}^{2} + {\left(x - 4\right)}^{2} / {b}^{2} = 1 \text{ [2]}$

Given that the major axis is 20 units, substitute 10 for "a" into equation [2]:

${\left(y - 2\right)}^{2} / {10}^{2} + {\left(x - 4\right)}^{2} / {b}^{2} = 1 \text{ [3]}$

Given that the minor axis is 6 units, substitute 3 for "b" into equation [3]:

${\left(y - 2\right)}^{2} / {10}^{2} + {\left(x - 4\right)}^{2} / {b}^{2} = 1 \text{ [4]}$

Equation [4] is specified equation. The following graphs is ellipse with the center and the end points of the major and minor axes plotted.