center, radius, graph, se, ge
c(-5,-1/2) tangent to x-axis

Given the problem where the center of the circle \( C \) is at \( (-5, -\frac{1}{2}) \), and the circle is tangent to the x-axis, we are asked to find the radius of the circle, and then use this information to graph the circle.

### Step 1: Finding the Radius
The radius \( r \) of the circle can be determined because the circle is tangent to the x-axis. The distance from the center of the circle to the x-axis is the radius. Since the center is at \( (-5, -\frac{1}{2}) \), the radius \( r \) is simply the distance from the point \( (-\frac{1}{2}) \) to the x-axis.

\[
r = \left| -\frac{1}{2} \right| = \frac{1}{2}
\]

### Step 2: Equation of the Circle
The general equation of a circle with center \( (h, k) \) and radius \( r \) is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting \( h = -5 \), \( k = -\frac{1}{2} \), and \( r = \frac{1}{2} \):

\[
(x + 5)^2 + \left( y + \frac{1}{2} \right)^2 = \left( \frac{1}{2} \right)^2
\]

Simplifying further:

\[
(x + 5)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{1}{4}
\]

### Step 3: Graphing the Circle
- **Center:** The center of the circle is at \( (-5, -\frac{1}{2}) \).
- **Radius:** The radius is \( \frac{1}{2} \).

The circle is tangent to the x-axis at the point \( (-5, 0) \), meaning it just touches the x-axis without crossing it.

### Summary:
- **Center:** \( (-5, -\frac{1}{2}) \)
- **Radius:** \( \frac{1}{2} \)
- **Equation of the Circle:** \( (x + 5)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{1}{4} \)

Would you like a graph of this circle, more details, or do you have any further questions?

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Here are some related questions to further your understanding:
1. How do you find the distance from a point to a line in the coordinate plane?
2. What happens to the radius if the circle is tangent to the y-axis instead?
3. How do you find the equation of a circle given two points it passes through?
4. What are the implications if the center of the circle lies on the x-axis?
5. Can a circle be tangent to both the x-axis and y-axis simultaneously? How?
6. What is the equation of a circle centered at the origin?
7. How does the equation of a circle change if the radius doubles?
8. What is the effect of translating a circle along the x or y axis?

**Tip:** Always check the sign and magnitude of the radius; it must always be a positive value.

Learn how to find the radius and graph of a circle tangent to the x-axis at the point (-5, -1/2). Includes step-by-step solution, circle equation, and graphical representation. Suitable for students in Grades 9-12.

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