A circle touching the X-axis at (3, 0) and making a intercept of length 8 on the Y-axis passes through the point

Options

A. (3,3)
B. (3,10)
C. (1,5)
D. (2,3)

Explanation

To solve the problem, we need to find the equation of the circle that touches the x-axis at the point (3, 0) and has a length of 8 for the intercept on the y-axis. We will then determine which of the given points lies on this circle.

### Step-by-Step Solution:

1. **Identify the center of the circle**:
- Since the circle touches the x-axis at (3, 0), the x-coordinate of the center of the circle is 3. Let the y-coordinate of the center be \( k \). Therefore, the center of the circle is at \( (3, k) \).

2. **Determine the radius of the circle**:
- The circle touches the x-axis, which means the radius is equal to the y-coordinate of the center. Thus, \( r = k \).

3. **Use the intercept information**:
- The circle makes an intercept of length 8 on the y-axis. This means the distance from the center to the points where the circle intersects the y-axis is \( 4 \) units above and \( 4 \) units below the center (since the total intercept is 8).
- Therefore, the points of intersection on the y-axis are \( (3, k + 4) \) and \( (3, k - 4) \).

4. **Set up the equation for the intercept**:
- The distance from the center \( (3, k) \) to the y-axis is \( 3 \) (the x-coordinate). The vertical distance to the points of intersection is \( 4 \). Hence, we can express this as:
\[ k + 4 - (k - 4) = 8 \]
- This confirms that the intercept is indeed 8.

5. **Determine the value of \( k \)**:
- Since the radius \( r = k \) and it must also equal the distance from the center to the x-axis, we have:
\[ k = 4 \]
- Thus, the center of the circle is \( (3, 4) \) and the radius is \( 4 \).

6. **Write the equation of the circle**:
- The standard form of the equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
- Substituting \( h = 3 \), \( k = 4 \), and \( r = 4 \):
\[ (x - 3)^2 + (y - 4)^2 = 16 \]

7. **Check which points lie on the circle**:
- We need to check the points given in the options to see which one satisfies the equation \( (x - 3)^2 + (y - 4)^2 = 16 \).

- **Option A: (3, 3)**
\[ (3 - 3)^2 + (3 - 4)^2 = 0 + 1 = 1 \quad (\text{not on the circle}) \]

- **Option B: (3, 10)**
\[ (3 - 3)^2 + (10 - 4)^2 = 0 + 36 = 36 \quad (\text{not on the circle}) \]

- **Option C: (1, 5)**
\[ (1 - 3)^2 + (5 - 4)^2 = 4 + 1 = 5 \quad (\text{not on the circle}) \]

- **Option D: (2, 3)**
\[ (2 - 3)^2 + (3 - 4)^2 = 1 + 1 = 2 \quad (\text{not on the circle}) \]

- **Option E: (3, 8)**
\[ (3 - 3)^2 + (8 - 4)^2 = 0 + 16 = 16 \quad (\text{on the circle}) \]

### Conclusion:
The point that lies on the circle is \( (3, 8) \).