Identify the vertices and the co-vertices in the graph. Is the ellipse horizontal or vertical?
(y+1)^2/25+(x+2)^2/9=1
Practice makes perfect
Let's start by graphing the ellipse. To do that, we will use the given information about the vertices and co-vertices.
As we can see on the graph, our ellipse is a vertical ellipse.
Let's recall the standard form of this type of ellipse, and highlight some of its important characteristics.
Vertical Ellipse
Standard Form
(y- k)^2/a^2+(x- h)^2/b^2=1, a and b positive, with a> b
Center
( h, k)
Vertices
( h, k± a)
Co-vertices
( h± b, k)
Length of Major Axis
2 a units
Length of Minor Axis
2b units
Foci
( h, k± c), c^2= a^2- b^2
From the graph we know that the x-coordinate of the vertices is - 1 and y-coordinate of the co-vertices is 6, so we can identify the center of the given ellipse.
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Vertex & Co-vertex & Center
( - 2,4) & (1, - 1) & ( - 2, - 1)
( - 2,- 6) & (- 5, - 1)
Because the center is ( - 2, - 1), we know that h= - 2 and k= - 1. Additionally, the length of the major axis is 2 a and the length of the minor axis is 2 b. Thus, a is the distance between the vertex and the center and b is the distance between the co-vertex and the center. From the graph we can tell that a= 5 and b= 3. Let's complete the equation!
(y-( - 1))^2/5^2+(x-( - 2))^2/3^2=1
⇕
(y+1)^2/25+(x+2)^2/9=1
