# How do you find the coordinates of the center, foci, the length of the major and minor axis given 27x^2+9y^2=81?

Apr 19, 2018

Center is at $\left(0 , 0\right)$ , major axis length is $6$ unit , minor axis length is $2 \sqrt{3}$ unit , focii at $\left(0 , - \sqrt{6}\right) , \left(0 , \sqrt{6}\right)$

#### Explanation:

$27 {x}^{2} + 9 {y}^{2} = 81 \mathmr{and} \frac{27 {x}^{2}}{81} + \frac{9 {y}^{2}}{81} = 1$ or

x ^2/3+ y^2/9= 1 or x ^2/ sqrt3^2+ y^2/3^2= 1 ; 3 > sqrt 3

This is standard equation of vertical ellipse with center at origin

(0,0) ; x^2/b^2+y^2/a^2=1 ; b = sqrt 3 ; a=3  . Major axis

length is $2 a = 6$ , Minor axis length is $2 b = 2 \sqrt{3}$

${c}^{2} = {a}^{2} - {b}^{2} = 9 - 3 = 6 \therefore c = \pm \sqrt{6}$

Focii at $\left(0 , - \sqrt{6}\right) , \left(0 , \sqrt{6}\right)$

graph{27x^2+9y^2=81 [-10, 10, -5, 5]}