# How do you write an equation of an ellipse given endpoints of major axis at (-11,5) and (7,5) and endpoints of the minor axis at (-2,9) and (-2,1)?

Dec 12, 2016

#### Explanation:

The endpoints, $\left(- 11 , 5\right) \mathmr{and} \left(7 , 5\right)$, of the major axis (where the x coordinate changes) have a general form of:

$\left(h - a , k\right) \mathmr{and} \left(h + a , k\right)$

This allows us to write the following equations:

$\text{[1] }$$k = 5$
$\text{[2] }$$h - a = - 11$
$\text{[3] }$$h + a = 7$

The endpoints, $\left(- 2 , 1\right) \mathmr{and} \left(- 2 , 9\right)$, of the minor axis (where the y coordinate changes) have a general form of:

$\left(h , k - b\right) \mathmr{and} \left(h , k + b\right)$

This allows us to write the following equations:

$\text{[4] }$$h = - 2$
$\text{[5] }$$k - b = 1$
$\text{[6] }$$k + b = 9$

Subtracting equation [2] from [3] gives us:

$2 a = 18$

$a = 9$

Subtracting equation [5] from [6] gives us:

$2 b = 8$

$b = 4$

All that remains, is to substitute these values into the general form for an ellipse with a horizontal major axis:

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

${\left(x - - 2\right)}^{2} / {9}^{2} + {\left(y - 5\right)}^{2} / {4}^{2} = 1$