A circle centered at (3, 2) with radius 5 intersects the x - axis at which of the following x - coordinates ?

Options

A. `2.39`
B. `4.58`
C. `7.58`
D. `8.00`

Explanation

To find the x-coordinates where the circle intersects the x-axis, we can follow these steps:

### Step 1: Write the equation of the circle
The standard equation of a circle with center \((x_1, y_1)\) and radius \(r\) is given by:
\[ (x - x_1)^2 + (y - y_1)^2 = r^2 \]
In this case, the center is \((3, 2)\) and the radius is \(5\). Therefore, we substitute these values into the equation:
\[ (x - 3)^2 + (y - 2)^2 = 5^2 \]
This simplifies to:
\[ (x - 3)^2 + (y - 2)^2 = 25 \]

### Step 2: Set \(y = 0\) to find the intersection with the x-axis
Since we want to find where the circle intersects the x-axis, we set \(y = 0\):
\[ (x - 3)^2 + (0 - 2)^2 = 25 \]
This simplifies to:
\[ (x - 3)^2 + 4 = 25 \]

### Step 3: Solve for \(x\)
Now, we can isolate \((x - 3)^2\):
\[ (x - 3)^2 = 25 - 4 \]
\[ (x - 3)^2 = 21 \]

### Step 4: Take the square root of both sides
Taking the square root of both sides gives:
\[ x - 3 = \pm \sqrt{21} \]

### Step 5: Solve for \(x\)
Now we can solve for \(x\):
\[ x = 3 \pm \sqrt{21} \]
Calculating the two possible values:
1. \(x = 3 + \sqrt{21}\)
2. \(x = 3 - \sqrt{21}\)

Calculating the numerical values:
- \(\sqrt{21} \approx 4.58\)
- Therefore, \(x \approx 3 + 4.58 \approx 7.58\)
- And \(x \approx 3 - 4.58 \approx -1.58\)

### Final Answer
The x-coordinates where the circle intersects the x-axis are approximately:
\[ x \approx -1.58 \quad \text{and} \quad x \approx 7.58 \]