Foci at (±2,0) and vertex at (−7/2,0)

Given the foci and vertex of the ellipse, we can determine its standard form. Here's how:

### Step 1: Identify the coordinates of the foci and the vertex

- **Foci**: \((\pm 2, 0)\) which means the foci are at \((2, 0)\) and \((-2, 0)\).
- **Vertex**: \(\left(-\frac{7}{2}, 0\right)\).

### Step 2: Determine the center of the ellipse
Since the foci are symmetric around the origin and lie on the \(x\)-axis, the center of the ellipse is at the origin \((0, 0)\).

### Step 3: Determine the distance from the center to the foci (\(c\))
The distance \(c\) from the center to either focus is:
\[
c = 2
\]

### Step 4: Determine the distance from the center to the vertex (\(a\))
The distance \(a\) from the center to the vertex \(\left(-\frac{7}{2}, 0\right)\) is:
\[
a = \left|0 - \left(-\frac{7}{2}\right)\right| = \frac{7}{2} = 3.5
\]

### Step 5: 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:
\[
2^2 = \left(\frac{7}{2}\right)^2 - b^2
\]
\[
4 = \frac{49}{4} - b^2
\]
\[
b^2 = \frac{49}{4} - \frac{16}{4} = \frac{33}{4}
\]
\[
b = \frac{\sqrt{33}}{2}
\]

### Step 6: Write the equation of the ellipse
Since the major axis is horizontal (along the \(x\)-axis), the standard form of the ellipse is:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Substituting \(a^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4}\) and \(b^2 = \frac{33}{4}\), the equation becomes:
\[
\frac{x^2}{\frac{49}{4}} + \frac{y^2}{\frac{33}{4}} = 1
\]
This simplifies to:
\[
\frac{4x^2}{49} + \frac{4y^2}{33} = 1
\]

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 center of an ellipse if the foci are not at the origin?
2. What happens to the shape of an ellipse as the distance between the foci increases?
3. How can you find the equation of an ellipse if you only know the length of the major and minor axes?
4. How does the equation of an ellipse change if the major axis is vertical instead of horizontal?
5. What is the geometric significance of the semi-minor axis in an ellipse?

**Tip:** When dealing with ellipses, always check whether the major axis is horizontal or vertical to use the correct form of the equation.

Learn how to determine the equation of an ellipse with foci at (±2, 0) and vertex at (−7/2, 0). This problem covers the standard form of an ellipse, including calculations for center, major and minor axes, and relationship between foci and vertex. Suitable for advanced high school students.

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