The given ellipse has foci at $(2, 0)$ and $(-2, 0)$, which suggests that the ellipse is centered at the origin and is horizontally oriented. Additionally, you mentioned that one of the vertices is at $(-\frac{7}{2}, 0)$. This gives us enough information to find the standard form of the equation of the ellipse.

Let's outline the process:

Step 1: Standard Form of an Ellipse

The standard form of the equation of an ellipse centered at the origin is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Where:

• $a$ is the semi-major axis (distance from the center to the vertex).
• $b$ is the semi-minor axis.
• $c$ is the distance from the center to the foci.

Step 2: Find $c$

From the information given, the foci are at $(2, 0)$ and $(-2, 0)$. The distance from the center to either focus is $c = 2$.

Step 3: Find $a$

The vertex is at $(-\frac{7}{2}, 0)$, which means the distance from the center to the vertex is $a = \frac{7}{2}$.

Step 4: Relationship Between $a$, $b$, and $c$

For an ellipse, the relationship between $a$, $b$, and $c$ is:

$c^2 = a^2 - b^2$

Substituting $c = 2$ and $a = \frac{7}{2}$:

$2^2 = \left( \frac{7}{2} \right)^2 - b^2$

$4 = \frac{49}{4} - b^2$

Step 5: Solve for $b^2$

Multiply the entire equation by 4 to eliminate the fraction:

$16 = 49 - 4b^2$

$4b^2 = 33$

$b^2 = \frac{33}{4}$

Step 6: Write the Equation

Now that we have $a^2 = \left( \frac{7}{2} \right)^2 = \frac{49}{4}$ and $b^2 = \frac{33}{4}$, we can plug these values into the standard form of the ellipse equation:

$\frac{x^2}{\frac{49}{4}} + \frac{y^2}{\frac{33}{4}} = 1$

Simplifying:

$\frac{4x^2}{49} + \frac{4y^2}{33} = 1$

This is the standard form of the ellipse.

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Would you like more details or have any questions? Here are five related questions to explore this further:

1. How do you determine whether an ellipse is horizontal or vertical?
2. How can you find the eccentricity of an ellipse?
3. What is the geometric significance of the foci in an ellipse?
4. How does the equation change if the ellipse is centered at a point other than the origin?
5. How do you calculate the area of an ellipse?

Tip: Always remember the relationship $c^2 = a^2 - b^2$ when dealing with ellipses.