# What is the equation of an ellipse ?

## Find the equation of an ellipse that passes through the points (2,3) and (1,-4)

Dec 24, 2016

${\left(y - - 4\right)}^{2} / {7}^{2} + {\left(x - 2\right)}^{2} / {1}^{2} = 1$

#### Explanation:

Excluding rotations, there are two general Cartesian forms for the equation of an ellipse:

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

and:

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

let k = -4, h = 2, and use equation [2}:

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

Use the point $\left(2 , 3\right)$

${\left(3 - - 4\right)}^{2} / {a}^{2} + {\left(2 - 2\right)}^{2} / {b}^{2} = 1$

$a = 7$

Use the point $\left(1 , - 4\right)$:

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

$b = 1$

The equation is:

${\left(y - - 4\right)}^{2} / {7}^{2} + {\left(x - 2\right)}^{2} / {1}^{2} = 1$