write the standard form of ellipse Vertices (– 7, – 3) and (13, – 3); and foci (– 5, – 3) and (11, – 3)

To write the standard form of the equation of an ellipse, we need to determine the center, the lengths of the semi-major and semi-minor axes, and the orientation of the ellipse. Let's break this down step by step.

### Step 1: Identify the center of the ellipse
The center \((h, k)\) of the ellipse is the midpoint of the line segment connecting the vertices.

Given vertices: \((-7, -3)\) and \((13, -3)\).

To find the midpoint:
\[
h = \frac{-7 + 13}{2} = \frac{6}{2} = 3
\]
\[
k = \frac{-3 + (-3)}{2} = -3
\]
So, the center is \((3, -3)\).

### Step 2: Determine the distance between the center and the vertices
The distance from the center to a vertex is the length of the semi-major axis, \(a\).

The distance between \((3, -3)\) and either of the vertices \((-7, -3)\) or \((13, -3)\) is:
\[
a = |3 - (-7)| = 10
\]

### Step 3: Determine the distance between the center and the foci
The distance from the center to a focus is the length of the focal distance, \(c\).

The distance between \((3, -3)\) and either of the foci \((-5, -3)\) or \((11, -3)\) is:
\[
c = |3 - (-5)| = 8
\]

### Step 4: Calculate \(b\) (semi-minor axis)
The relationship between \(a\), \(b\), and \(c\) for an ellipse is given by:
\[
c^2 = a^2 - b^2
\]
Substituting the known values:
\[
8^2 = 10^2 - b^2
\]
\[
64 = 100 - b^2
\]
\[
b^2 = 36 \quad \text{so} \quad b = 6
\]

### Step 5: Write the equation of the ellipse
Since the major axis is horizontal (the vertices and foci have the same \(y\)-coordinate), the standard form of the ellipse is:
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
Substituting the values \(h = 3\), \(k = -3\), \(a^2 = 100\), and \(b^2 = 36\), the equation becomes:
\[
\frac{(x - 3)^2}{100} + \frac{(y + 3)^2}{36} = 1
\]

This is the standard form of the ellipse.

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. How do you determine the length of the semi-major axis from the vertices?
2. What is the significance of the foci in an ellipse?
3. How would the equation change if the ellipse were vertical?
4. How can you derive the equation of an ellipse given only the foci and one vertex?
5. What would happen to the ellipse if the distance between the foci increased?

**Tip:** Remember that the distance between the foci determines how "stretched" the ellipse is; as the foci get closer to the center, the ellipse becomes more circular.

Understand how to write the standard form of an ellipse given vertices and foci. This detailed solution covers finding the center, calculating semi-major and semi-minor axes, and deriving the equation. Suitable for advanced high school students studying conic sections.

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