For vertical ellipses the foci are (0,± c). For horizontal ellipses the foci are (± c,0). You can find c 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.
(0,± 1)
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.
Note that 2sqrt(6) ≈ 4.9. Therefore, since 5> 2sqrt(6), the given equation represents a vertical ellipse.
x^2/(2 sqrt(6))^2+y^2/5^2=1
Type of Ellipse
Vertical
Vertices
(0,± 5)
Co-vertices
(± 2sqrt(6),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 5 for a and sqrt(6) 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 we know that c is a positive number. We already found that our equation represents a vertical ellipse. For vertical ellipses the foci are (0,± c). Since c=1, the foci of the given ellipse are (0,1) and (0,- 1).
