Let `S = {(2alpha+3i)/(2alpha-3i):alpha in R}`. All the points of S lie on a circle of radius ____________.

Answer

To solve the problem, we need to analyze the complex number \( S = \frac{2\alpha + 3i}{2\alpha - 3i} \) where \( \alpha \) is a real number. We will find the radius of the circle on which all points of \( S \) lie.

### Step-by-Step Solution:

1. **Express S in terms of x and y**:
We start with the expression for \( S \):
\[ S = \frac{2\alpha + 3i}{2\alpha - 3i} \]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ S = \frac{(2\alpha + 3i)(2\alpha + 3i)}{(2\alpha - 3i)(2\alpha + 3i)} \]

2. **Calculate the denominator**:
The denominator becomes:
\[ (2\alpha - 3i)(2\alpha + 3i) = (2\alpha)^2 - (3i)^2 = 4\alpha^2 + 9 \]

3. **Calculate the numerator**:
The numerator becomes:
\[ (2\alpha + 3i)(2\alpha + 3i) = (2\alpha)^2 + 2(2\alpha)(3i) + (3i)^2 = 4\alpha^2 + 12\alpha i - 9 \]

4. **Combine the results**:
Thus, we have:
\[ S = \frac{4\alpha^2 - 9 + 12\alpha i}{4\alpha^2 + 9} \]
This can be rewritten as:
\[ S = \frac{4\alpha^2 - 9}{4\alpha^2 + 9} + i \frac{12\alpha}{4\alpha^2 + 9} \]
Let \( x = \frac{4\alpha^2 - 9}{4\alpha^2 + 9} \) and \( y = \frac{12\alpha}{4\alpha^2 + 9} \).

5. **Find the equation of the circle**:
To find the radius of the circle, we need to compute \( x^2 + y^2 \):
\[ x^2 + y^2 = \left(\frac{4\alpha^2 - 9}{4\alpha^2 + 9}\right)^2 + \left(\frac{12\alpha}{4\alpha^2 + 9}\right)^2 \]
This simplifies to:
\[ = \frac{(4\alpha^2 - 9)^2 + (12\alpha)^2}{(4\alpha^2 + 9)^2} \]

6. **Simplify the numerator**:
Expanding the numerator:
\[ (4\alpha^2 - 9)^2 + (12\alpha)^2 = 16\alpha^4 - 72\alpha^2 + 81 + 144\alpha^2 = 16\alpha^4 + 72\alpha^2 + 81 \]

7. **Final expression**:
Thus, we have:
\[ x^2 + y^2 = \frac{16\alpha^4 + 72\alpha^2 + 81}{(4\alpha^2 + 9)^2} \]

8. **Recognize the form**:
Notice that the numerator can be factored:
\[ 16\alpha^4 + 72\alpha^2 + 81 = (4\alpha^2 + 9)^2 \]
Therefore, we have:
\[ x^2 + y^2 = \frac{(4\alpha^2 + 9)^2}{(4\alpha^2 + 9)^2} = 1 \]

9. **Conclusion**:
Since \( x^2 + y^2 = 1 \), this indicates that all points \( S \) lie on a circle with a radius of 1.

### Final Answer:
The radius of the circle is **1**.