Identify the type of ellipse. Horizontal ellipses fit the equation (x-h)^2a^2+ (y-k)^2b^2=1, and vertical ellipses fit the equation (x-h)^2b^2+ (y-k)^2a^2=1. In both cases, a and b are positive numbers such that a>b.
Center: (- 7, - 1) Vertices: (8,- 1) and (- 22, - 1) Foci: (2,- 1) and (- 16,- 1)
Practice makes perfect
We want to identify the center, vertices, and foci of the ellipse. To do so, we will rewrite the given equation just a little bit.
Notice that the denominator of the expression containing the x-variable is greater than the denominator of the expression that contains the y-variable. Therefore, we have a horizontal ellipse. Let's recall the main characteristics of this type of ellipse.
Horizontal Ellipse
Standard Form Equation
(x- h)^2/a^2+(y- k)^2/b^2=1
Center
( h, k)
Vertices
( h± a, k)
Co-vertices
( h, k ± b)
Foci
( h± c, k)
a,b,c relationship, a>b>0
c^2= a^2- b^2
Let's consider our equation one more time.
(x-( - 7))^2/15^2+(y-( - 1))^2/12^2=1
We can see that a= 15, b= 12, h= - 7, and k= - 1. The only value we do not know is c. Let's find it!
Note that we only took the principal root, because to find the foci we will add and subtract the value of c. Therefore, its sign is irrelevant. Finally, we can write the desired information.
