To determine the equation of the ellipse, we need to utilize the information given:
1. Vertices and Co-vertices:
- The vertex is at [tex]\((3, 0)\)[/tex].
- The co-vertex is at [tex]\((0, -1)\)[/tex].
2. Center:
- The center is at the origin [tex]\((0, 0)\)[/tex].
Using this information, we'll identify the parameters of the ellipse:
### Step 1: Identify the distances [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
- Distance from the center to the vertex at (3, 0): This distance is referred to as [tex]\(a\)[/tex], which represents the semi-major axis.
[tex]\[
a = 3
\][/tex]
- Distance from the center to the co-vertex at (0, -1): This distance is referred to as [tex]\(b\)[/tex], which represents the semi-minor axis.
[tex]\[
b = 1
\][/tex]
### Step 2: Form the equation of the ellipse
The standard form of the equation of an ellipse with the major axis along the x-axis and centered at the origin is:
[tex]\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\][/tex]
### Step 3: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the equation
- [tex]\(a = 3\)[/tex], so [tex]\(a^2 = 3^2 = 9\)[/tex]
- [tex]\(b = 1\)[/tex], so [tex]\(b^2 = 1^2 = 1\)[/tex]
Substitute these values into the standard form equation:
[tex]\[
\frac{x^2}{9} + \frac{y^2}{1} = 1
\][/tex]
### Step 4: Simplify the equation
[tex]\[
\frac{x^2}{9} + y^2 = 1
\][/tex]
Thus, the correct equation of the ellipse is:
[tex]\[
\boxed{\frac{x^2}{9} + y^2 = 1}
\][/tex]
