To find the standard form of the equation of the ellipse, we start by identifying the details provided:
- Foci: (0, 6) and (0, -6)
- Vertices: (0, 8) and (0, -8)
The general standard form of the equation of an ellipse centered at the origin with a vertical major axis is:
\(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\
where:
- \(a\) is the distance from the center to a vertex,
- \(b\) is the distance from the center to the co-vertices,
- The distance between the center and each focus is denoted as \(c\), which can be found using the relation \(c^2 = a^2 – b^2\).
In this case, we can determine:
- The vertices indicate that \(a = 8\).
- The foci indicate that \(c = 6\).
Using the relationship \(c^2 = a^2 – b^2\), we can substitute our values:
\(c^2 = 6^2 = 36\)
\(a^2 = 8^2 = 64\)
Now, substitute into the relationship:
\(36 = 64 – b^2\
Solving for \(b^2\):
\(b^2 = 64 – 36 = 28\
Now we have:
- \(a^2 = 64\
- \(b^2 = 28\
Finally, we can write the equation of the ellipse in standard form as:
\(\frac{x^2}{28} + \frac{y^2}{64} = 1\
This is the standard form of the equation of the ellipse with the given characteristics.