To write the equation of an ellipse, we need to determine a few key pieces of information based on the given characteristics.

Step 1: Identify the center

The vertices are $(4, 3)$ and $(4, -9)$. Since the vertices are aligned vertically, the center of the ellipse is the midpoint of the line segment connecting the two vertices. The midpoint can be calculated as: $\text{Center} = \left( \frac{4 + 4}{2}, \frac{3 + (-9)}{2} \right) = (4, -3)$

Thus, the center of the ellipse is $(4, -3)$.

Step 2: Determine the value of $a$ (the semi-major axis)

The distance between the two vertices is the total length of the major axis, which is 12 (since $3 - (-9) = 12$). The semi-major axis $a$ is half of this length: $a = \frac{12}{2} = 6$

Since the major axis is vertical, this means the term involving $a^2$ will be under the $(y - k)^2$ term in the equation.

Step 3: Determine the value of $b$ (the semi-minor axis)

The length of the minor axis is given as 8. The semi-minor axis $b$ is half of this length: $b = \frac{8}{2} = 4$

Step 4: Write the equation of the ellipse

The general form of the equation of an ellipse centered at $(h, k)$ is: $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ Since the center is $(4, -3)$, $a = 6$, and $b = 4$, the equation becomes: $\frac{(x - 4)^2}{4^2} + \frac{(y + 3)^2}{6^2} = 1$ Simplifying: $\frac{(x - 4)^2}{16} + \frac{(y + 3)^2}{36} = 1$

Final Answer:

The equation of the ellipse is: $\frac{(x - 4)^2}{16} + \frac{(y + 3)^2}{36} = 1$

---

Would you like further details on this process or have any questions?
Here are some related questions for practice:

1. How would the equation change if the major axis were horizontal instead of vertical?
2. What is the eccentricity of the ellipse described by the equation?
3. How do you find the foci of this ellipse?
4. What are the coordinates of the endpoints of the minor axis?
5. How do you rewrite the equation of an ellipse if it is not centered at the origin?

Tip: The sum of the distances from any point on an ellipse to the two foci is always constant and equals the length of the major axis.