Start by identifying whether you have a horizontal or a vertical ellipse.
Equation: (x-2)^29+ (y-5)^24=1 Graph:
Practice makes perfect
Let's start by recalling the standard form for horizontal and vertical ellipses and highlight some of their important characteristics.
Horizontal Ellipse
Vertical Ellipse
Standard Form
(x- h)^2/a^2+(y- k)^2/b^2=1, a and b positive, with a> b
(x- h)^2/b^2+(y- k)^2/a^2=1, a and b positive, with a> b
Center
( h, k)
( h, k)
Vertices
( h± a, k)
( h, k± a)
Co-Vertices
( h, k± b)
( h± b, k)
Length of Major Axis
2 a units
2 a units
Length of Minor Axis
2 b units
2 b units
Foci
( h± c, k), c^2= a^2- b^2
( h, k± c), c^2= a^2- b^2
We are given the center, vertices, and co-vertices of an ellipse, and we want to write its equation. Note that the center has the same x-coordinate as the co-vertices and the same y-coordinate as the vertices. According to our table, this corresponds to a horizontal ellipse.
ccc
Center & Vertices & Co-vertices
( 2, 5)& (5, 5)and(- 1, 5)& ( 2,3)and ( 2,7)
We already know that h= 2 and that k= 5. Let's use one of the vertices and the center to find the positive value of a.
Following the same procedure, we will use one of the co-vertices and the center to find the positive value of b.
Knowing that a= 3 and b= 2, we now have everything we need to write the equation of the ellipse.
(x- 2)^2/3^2+(y- 5)^2/2^2=1
⇕
(x-2)^2/9+(y-5)^2/4=1
Now, let's draw our ellipse. To do so, we will first isolate the y-variable.
