# How do you write an equation of an ellipse in standard form given center is at origin, major axis is 18 minor axis is 6?

Nov 15, 2015

${x}^{2} / 81 + {y}^{2} / 9 = 1$ if the major axis is horizontal
${x}^{2} / 9 + {y}^{2} / 81 = 1$ if the major axis is vertical

#### Explanation:

$C : \left(0 , 0\right)$

$M = 18 = 2 a \implies a = 9$

$m = 6 = 2 b \implies b = 3$

If the major axis is horizontal, the standard equation of the ellipse is

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

$\implies {\left(x - 0\right)}^{2} / {9}^{2} + {\left(y - 0\right)}^{2} / {3}^{2} = 1$

$\implies {x}^{2} / 81 + {y}^{2} / 9 = 1$

If the major axis is vertical, the standard equation of the ellipse is

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

$\implies {\left(x - 0\right)}^{2} / {3}^{2} + {\left(y - 0\right)}^{2} / {9}^{2} = 1$

$\implies {x}^{2} / 9 + {y}^{2} / 81 = 1$