For vertical ellipses the foci are (0,± c). For horizontal ellipses the foci are (± c,0). The value of c can be found by solving the equation c^2=a^2-b^2, where a and b are the absolute values of the nonzero coordinate of the vertices and co-vertices, respectively.
Foci: (0,± sqrt(6)) Graph:
Practice makes perfect
Let's start by recalling the general equation of horizontal and vertical ellipses.
ccc
Horizontal Ellipse& & Vertical Ellipse [0.5em]
x^2/a^2+y^2/b^2=1 & & x^2/b^2+y^2/a^2=1
For both equations above a and b are positive numbers, with a>b. Moreover, ± a and ± b are the nonzero coordinates of the vertices and co-vertices, respectively.
To find the foci of the ellipse whose equation is given, we will rewrite the equation to match one of the above formats.
Since sqrt(3) ≈ 1.7 and therefore 3>sqrt(3), the given equation represents a vertical ellipse.
Equation
3x^2+y^2=9 ⇕ x^2/( sqrt(3))^2+y^2/3^2=1
Type of Ellipse
Vertical
Vertices
(0,± 3)
Co-vertices
(± sqrt(3),0)
Foci
(0,± c)
The value of c, which is the absolute value of the nonzero coordinate of the foci, can be expressed in terms of a and b.
c^2=a^2-b^2
Let's substitute 3 for a and sqrt(3) for b in the above equation so that we can solve for c.
Note that in the last step we took the principal root, because c is a positive number. The foci of the given ellipse are (0,sqrt(6)) and (0,- sqrt(6)).
Finally, let's graph our ellipse.
