What's is the equation if the Vertices(-5,-2)  and (11, -2 ) minor axis length  is 12

To find the equation of the ellipse with the given vertices and minor axis length, we need to use the standard form of the ellipse equation:

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

where:
- \((h, k)\) is the center of the ellipse,
- \(2a\) is the length of the major axis,
- \(2b\) is the length of the minor axis.

### Step 1: Find the center \((h, k)\)
The vertices \((-5, -2)\) and \((11, -2)\) are given. The center of the ellipse is the midpoint of these two points.

\[
h = \frac{-5 + 11}{2} = \frac{6}{2} = 3, \quad k = \frac{-2 + (-2)}{2} = -2
\]

So, the center of the ellipse is \((3, -2)\).

### Step 2: Find the lengths of the semi-major axis \(a\) and the semi-minor axis \(b\)
- The distance between the vertices is \(2a\).
  
  \[
  2a = \text{distance between } (-5, -2) \text{ and } (11, -2) = \sqrt{(11 - (-5))^2 + (-2 - (-2))^2} = \sqrt{16^2} = 16
  \]

  Therefore, \(a = 8\).

- The length of the minor axis is given as \(12\), so \(2b = 12\).

  Therefore, \(b = 6\).

### Step 3: Write the equation of the ellipse
Since the major axis is horizontal (as the vertices vary in \(x\)), the equation is:

\[
\frac{(x-3)^2}{8^2} + \frac{(y+2)^2}{6^2} = 1
\]

Simplifying, we get:

\[
\frac{(x-3)^2}{64} + \frac{(y+2)^2}{36} = 1
\]

This is the equation of the ellipse.

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Would you like more details on any part of this solution? Here are some related questions you might find interesting:

1. How do you find the foci of an ellipse?
2. What is the relationship between the major axis and the minor axis?
3. How does the equation of an ellipse change if it is rotated?
4. What is the significance of the eccentricity in an ellipse?
5. How do you determine the area of an ellipse?

**Tip:** Always double-check the orientation of the major and minor axes before writing the equation of an ellipse!

Learn how to find the equation of an ellipse with vertices (-5,-2) and (11,-2) and a minor axis length of 12. This problem involves using the standard form of the ellipse equation and calculating the center, semi-major axis, and semi-minor axis. Suitable for high school students studying geometry and conic sections.

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