To find the standard form of the equation of an ellipse that has a vertex at (0,6), a co-vertex at (1,0), and a center at the origin (0,0), follow these steps:
1. Identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- The vertex (0,6) indicates that the distance from the center to a vertex along the y-axis is [tex]\(a = 6\)[/tex].
- The co-vertex (1,0) indicates that the distance from the center to a co-vertex along the x-axis is [tex]\(b = 1\)[/tex].
2. Calculate [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
- [tex]\(a^2 = 6^2 = 36\)[/tex]
- [tex]\(b^2 = 1^2 = 1\)[/tex]
3. Write the standard form of the ellipse:
- For an ellipse with a vertical major axis centered at the origin, the standard form of the equation is:
[tex]\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\][/tex]
4. Plug the values of [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex] into the equation:
- Substituting [tex]\(a^2 = 36\)[/tex] and [tex]\(b^2 = 1\)[/tex], the equation becomes:
[tex]\[
\frac{x^2}{1} + \frac{y^2}{36} = 1
\][/tex]
So, the correct equation in standard form is:
[tex]\[
\boxed{\frac{x^2}{1} + \frac{y^2}{36} = 1}
\][/tex]
