How do you write an equation for an ellipse given endpoints of the major axis at (2,2) and (2,-10) and endpoints of the minor axis at (0,-4) and (4,-4)?

Jan 16, 2017

${\left(x - 2\right)}^{2} / 16 + {\left(y + 3\right)}^{2} / 36 = 1$

Explanation:

Find the center of the ellipse

$\left(x , y\right) = \frac{2 + 2}{2} , \frac{2 + \left(- 10\right)}{2} = 2 , - 5$

The Ellipse center is$\left(2 , - 3\right)$; the ellipse is vertical; hence its equation must be

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

$\left(h , k\right) = \left(2 , - 3\right)$

$a$ is the major axis

$b$ is the minor axis

$a = 6$
$b = 4$

Then the equation is -

${\left(x - 2\right)}^{2} / {4}^{2} + {\left(y + 3\right)}^{2} / {6}^{2} = 1$
${\left(x - 2\right)}^{2} / 16 + {\left(y + 3\right)}^{2} / 36 = 1$