A circle is drawn whose centre is on the x - axis and it touches the y - axis. If no part of the circle lies outside the parabola `y^(2)=8x`, then the maximum possible radius of the circle is

Answer

To solve the problem, we need to determine the maximum possible radius of a circle whose center is on the x-axis, touches the y-axis, and is entirely contained within the parabola defined by \( y^2 = 8x \).

### Step-by-Step Solution:

1. **Understanding the Circle's Position**:
- Let the center of the circle be at the point \( (h, 0) \) on the x-axis.
- Since the circle touches the y-axis, the radius \( r \) of the circle is equal to \( h \). Therefore, we can express the circle's equation as:
\[ (x - h)^2 + y^2 = h^2 \]

2. **Finding the Circle's Boundary**:
- The circle will extend from \( (h - h, 0) = (0, 0) \) to \( (h + h, 0) = (2h, 0) \) along the x-axis and will have a maximum height of \( r = h \) above and below the x-axis.

3. **Analyzing the Parabola**:
- The parabola \( y^2 = 8x \) opens to the right. To find the intersection points of the circle and the parabola, we need to substitute \( y^2 = 8x \) into the equation of the circle.

4. **Substituting for y**:
- From the parabola, we have \( x = \frac{y^2}{8} \).
- Substitute this into the circle's equation:
\[ \left(\frac{y^2}{8} - h\right)^2 + y^2 = h^2 \]

5. **Simplifying the Equation**:
- Expanding the equation gives:
\[ \left(\frac{y^2 - 8h}{8}\right)^2 + y^2 = h^2 \]
- This leads to a quartic equation in \( y \). However, we are interested in ensuring that the circle does not extend outside the parabola.

6. **Condition for Containment**:
- For the circle to be entirely within the parabola, the maximum height of the circle (which is \( h \)) must be less than or equal to the y-coordinate of the parabola at the point where the circle touches the y-axis.
- The parabola's value at \( x = h \) is:
\[ y^2 = 8h \implies y = \sqrt{8h} \]
- For the circle to be contained, we need:
\[ h \leq \sqrt{8h} \]

7. **Solving the Inequality**:
- Squaring both sides gives:
\[ h^2 \leq 8h \]
- Rearranging this leads to:
\[ h^2 - 8h \leq 0 \]
- Factoring gives:
\[ h(h - 8) \leq 0 \]
- This inequality holds for \( 0 \leq h \leq 8 \).

8. **Finding Maximum Radius**:
- The maximum possible radius \( r \) of the circle is equal to \( h \), and thus the maximum value of \( h \) is 8. Therefore, the maximum radius of the circle is:
\[ r = 4 \]

### Conclusion:
The maximum possible radius of the circle is \( \boxed{4} \).