How do you write an equation for a ellipse with center (5, -4), vertical major axis of length 12, and minor axis of length 8?

Jun 5, 2016

${\left(x - 5\right)}^{2} / 36 + {\left(y + 4\right)}^{2} / 16 = 1$

Explanation:

Standard equations of an ellipse

Major axis is horizontal
${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$

Major axis is vertical
${\left(x - h\right)}^{2} / {b}^{2} + {\left(y - k\right)}^{2} / {a}^{2} = 1$

where:
Center: $\left(h , k\right)$
Major axis: $2 a$
minor axis: $2 b$

In the given
Center: $\left(5 , - 4\right)$
Major axis: $12 = 2 a \implies a = 6$
minor axis: $8 = 2 b \implies b = 4$

We are dealing with an ellipse with a vertical major axis, so we should use the second form of the standard equation

${\left(x - 5\right)}^{2} / {6}^{2} + {\left(y + 4\right)}^{2} / {4}^{2} = 1$

$\implies {\left(x - 5\right)}^{2} / 36 + {\left(y + 4\right)}^{2} / 16 = 1$