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: (± 2sqrt(3),0) 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 4>2, the given equation represents a horizontal ellipse.
Equation
x^2+4y^2=16 ⇕ x^2/4^2+y^2/2^2=1
Type of Ellipse
Horizontal
Vertices
(± 4,0)
Co-vertices
(0,± 2)
Foci
(± c,0)
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 4 for a and 2 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 (2sqrt(3),0) and (- 2sqrt(3),0).
Finally, let's graph our ellipse.
