To find the standard form of the equation of an ellipse given its foci and vertices, we can follow these steps:
Step 1: Identify the Foci and Vertices
The foci of the ellipse are located at (0, 2) and (0, -2). The vertices are at (0, 8) and (0, -8). The foci are vertical and centered along the y-axis, which indicates that the major axis is also vertical.
Step 2: Determine the Center of the Ellipse
The center of the ellipse is the midpoint between the vertices. The y-coordinates of the vertices are -8 and 8. Therefore, the center (h, k) is:
h = 0, k = (8 + (-8))/2 = 0.
So, the center of the ellipse is (0, 0).
Step 3: Calculate the Lengths of the Major and Minor Axes
The distance from the center to each vertex gives us the semi-major axis length a.
Since the distance from the center (0, 0) to a vertex (0, 8) is:
a = 8.
Next, we need to find the distance from the center to each focus, which gives us the focal distance c.
The distance from the center (0, 0) to a focus (0, 2) is:
c = 2.
Step 4: Calculate the Semi-Minor Axis Length
The relationship for an ellipse is given by the equation:
c² = a² – b²,
where b is the semi-minor axis length.
Plugging in the values:
2² = 8² – b²
4 = 64 – b²
Now, solving for b²:
b² = 64 – 4 = 60
Thus, b = √60 = 2√15.
Step 5: Write the Standard Form of the Ellipse Equation
The standard form of the ellipse with a vertical major axis is:
\( \frac{(x – h)^{2}}{b^{2}} + \frac{(y – k)^{2}}{a^{2}} = 1 \)
With our center (h, k) = (0, 0), a = 8, and b = 2√15, we substitute:
\( \frac{x^{2}}{60} + \frac{y^{2}}{64} = 1 \)
Conclusion
The standard form of the equation of the ellipse with foci at (0, 2) and (0, -2) and vertices at (0, 8) and (0, -8) is:
\( \frac{x^{2}}{60} + \frac{y^{2}}{64} = 1 \)
This equation describes an ellipse centered at the origin, with a vertical major axis.