V(-2,2) and (-2,8) and length of minor axis is 6

You're describing an ellipse, where the two points are likely the **vertices** along either the major or minor axis, and the length of the minor axis is provided as 6.

Let's break down the problem step by step:

1. **Vertices**: You provided two points, \( V(-2, 2) \) and \( (-2, 8) \). These points have the same \(x\)-coordinate, meaning they lie vertically along the **y-axis**. This suggests that these are the endpoints of the **major axis** (because the major axis is usually longer than the minor axis, and the points are further apart vertically).

2. **Length of Major Axis**: The distance between these two points can be calculated as the difference in the \(y\)-coordinates, which is:
   \[
   8 - 2 = 6
   \]
   So, the length of the **major axis** is 6.

3. **Center of the Ellipse**: The center is located at the midpoint of the major axis. The midpoint between \( V(-2, 2) \) and \( (-2, 8) \) is:
   \[
   \left( \frac{-2 + (-2)}{2}, \frac{2 + 8}{2} \right) = (-2, 5)
   \]
   So, the center of the ellipse is at \( (-2, 5) \).

4. **Length of Minor Axis**: It's given that the length of the **minor axis** is 6.

### Equation of the Ellipse
The standard form for the equation of an ellipse centered at \( (h, k) \) is:
\[
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1
\]
where:
- \( (h, k) \) is the center,
- \( a \) is half the length of the major axis,
- \( b \) is half the length of the minor axis.

In this case:
- The center is \( (-2, 5) \),
- The length of the major axis is 6, so \( a = \frac{6}{2} = 3 \),
- The length of the minor axis is 6, so \( b = \frac{6}{2} = 3 \).

Thus, the equation of the ellipse is:
\[
\frac{(x + 2)^2}{3^2} + \frac{(y - 5)^2}{3^2} = 1
\]
or simplified:
\[
\frac{(x + 2)^2}{9} + \frac{(y - 5)^2}{9} = 1
\]

This is the equation of the ellipse.

### Do you need any further details or clarifications?

Here are 5 relative questions to explore further:
1. How do you find the foci of an ellipse?
2. What happens if the lengths of the axes are changed?
3. Can you calculate the eccentricity of this ellipse?
4. How would the equation of an ellipse change if the center was moved to the origin?
5. What is the general process for deriving the equation of an ellipse from geometric data?

**Tip**: Always verify which axis (major or minor) the given points represent, as this affects how the equation is constructed!

Learn how to derive the equation of an ellipse with vertices at V(-2, 2) and (-2, 8) and a minor axis length of 6. This problem covers key geometric concepts, including the formula for an ellipse, and how to calculate the center, axes, and equation. Suitable for students in Grades 10-12.

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