The question asks for the standard equation of an ellipse with the given features:

• Vertices at $(4, -2)$ and $(4, 8)$
• Minor axis of length 6.

Step 1: Identifying the center

The vertices have the same x-coordinate, so the major axis is vertical. The center of the ellipse is the midpoint between the vertices:

$\text{Center} = \left(\frac{4 + 4}{2}, \frac{-2 + 8}{2}\right) = (4, 3)$

Step 2: Length of the semi-major axis

The distance between the vertices is the length of the major axis. The formula for distance between two points is:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Since the x-coordinates are the same, the distance is simply the difference in y-coordinates:

$\text{Major Axis Length} = |8 - (-2)| = 10$

The semi-major axis, $a$, is half the length of the major axis:

$a = \frac{10}{2} = 5$

Step 3: Length of the semi-minor axis

The minor axis is given as 6, so the semi-minor axis, $b$, is:

$b = \frac{6}{2} = 3$

Step 4: Standard form of the ellipse equation

For an ellipse centered at $(h, k)$ with a vertical major axis, the equation is:

$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

Substituting the center $(h, k) = (4, 3)$, $a = 5$, and $b = 3$:

$\frac{(x - 4)^2}{3^2} + \frac{(y - 3)^2}{5^2} = 1$

Simplifying:

$\frac{(x - 4)^2}{9} + \frac{(y - 3)^2}{25} = 1$

Final Answer:

The standard equation of the ellipse is:

$\frac{(x - 4)^2}{9} + \frac{(y - 3)^2}{25} = 1$

Would you like more details on any part of this solution? Here are some related questions to further explore the concept:

1. How do you calculate the distance between two points in a plane?
2. How do you identify the orientation of the ellipse from its vertices?
3. What is the difference between the major and minor axes of an ellipse?
4. How would the equation change if the major axis were horizontal instead of vertical?
5. What role do the values of $a$ and $b$ play in the shape of the ellipse?

Tip: In an ellipse, the larger value between $a$ and $b$ corresponds to the major axis, and the smaller value corresponds to the minor axis.