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We are given that the foci of the ellipse are located at \((\pm 4, 0)\), which indicates the ellipse is horizontally oriented. Additionally, we are told that the point \(P(4, 6)\) lies on the ellipse.

### Step 1: General Form of an Ellipse Equation
For an ellipse centered at \((0, 0)\) with horizontal major axis, the equation is:

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

Where:
- \(a\) is the semi-major axis length.
- \(b\) is the semi-minor axis length.
- The foci are at \((\pm c, 0)\), where \(c\) is the distance from the center to each focus.

### Step 2: Relationship Between \(a\), \(b\), and \(c\)
The relationship between \(a\), \(b\), and \(c\) for an ellipse is:

\[
c^2 = a^2 - b^2
\]

We are given that the foci are at \((\pm 4, 0)\), meaning \(c = 4\). So,

\[
c^2 = 16
\]

### Step 3: Using the Point \(P(4, 6)\)
Since \(P(4, 6)\) lies on the ellipse, it satisfies the equation of the ellipse. So, we substitute \(x = 4\) and \(y = 6\) into the equation:

\[
\frac{4^2}{a^2} + \frac{6^2}{b^2} = 1
\]

This gives:

\[
\frac{16}{a^2} + \frac{36}{b^2} = 1
\]

### Step 4: Finding the Vertices
To find the distance from the center of the ellipse to one of its vertices, we need to find \(a\). The vertices are located at \((\pm a, 0)\). Since we already know \(c = 4\), we can use the equations to determine \(a\) and solve for the distance to the vertices.

I will solve for \(a\) and \(b\).From the solutions, the relevant real values for \(a\) and \(b\) are:

\[
a = 8, \quad b = 4\sqrt{3}
\]

### Step 5: Distance to One of the Vertices
The vertices of the ellipse are located at \((\pm a, 0)\). Therefore, the distance from the center of the ellipse to one of its vertices is simply \(a = 8\).

### Final Answer:
The distance from the center of the ellipse to one of its vertices is **8 units**.

---

Would you like further details or clarification? Here are some related questions for deeper understanding:

1. What is the significance of the values \(a\), \(b\), and \(c\) in the context of an ellipse?
2. How do we derive the general equation of an ellipse from its foci?
3. What happens to the shape of the ellipse as \(a\) and \(b\) vary?
4. How does the formula \(c^2 = a^2 - b^2\) help in identifying the ellipse's properties?
5. How do we verify that a given point lies on an ellipse?

**Tip:** In an ellipse, the larger the value of \(a\) compared to \(b\), the more "stretched" it becomes horizontally.

The foci of an ellipse are at (±4, 0). Find the distance from the center of the ellipse to one of its vertices if P(4, 6) is a point on the graph.

Learn how to find the distance from the center of an ellipse to its vertex, given the foci and a point on the ellipse. This detailed solution includes the equation of an ellipse, the relationship between its axes, and step-by-step calculations. Suitable for Grades 9-12.

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