Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.
1. **Define the problem geometrically**:
   - Let \( k \) be a circle with center \( O \) and radius \( r \).
   - Let \( A \) be a fixed point on \( k \).
   - Let \( B \) be any point on \( k \).
   - Let \( M \) be a point such that \( \frac{AM}{AB} = m \) and \( \angle BAM = \alpha \).

2. **Express the coordinates of points**:
   - Assume \( A \) is at coordinates \( (r, 0) \).
   - Let \( B \) be at coordinates \( (r \cos \theta, r \sin \theta) \) for some angle \( \theta \).

3. **Use the given ratio**:
   - The distance \( AB \) is given by the Euclidean distance formula:
     \[
     AB = \sqrt{(r \cos \theta - r)^2 + (r \sin \theta - 0)^2} = r \sqrt{2 - 2 \cos \theta} = 2r \sin \left( \frac{\theta}{2} \right)
     \]
   - Since \( \frac{AM}{AB} = m \), we have:
     \[
     AM = m \cdot AB = 2mr \sin \left( \frac{\theta}{2} \right)
     \]

4. **Determine the coordinates of \( M \)**:
   - Let \( M \) be at coordinates \( (x, y) \).
   - Using the angle \( \alpha \), we can write the coordinates of \( M \) in terms of \( A \) and \( B \):
     \[
     x = r + AM \cos \alpha = r + 2mr \sin \left( \frac{\theta}{2} \right) \cos \alpha
     \]
     \[
     y = AM \sin \alpha = 2mr \sin \left( \frac{\theta}{2} \right) \sin \alpha
     \]

5. **Simplify the coordinates**:
   - The coordinates of \( M \) are:
     \[
     x = r + 2mr \sin \left( \frac{\theta}{2} \right) \cos \alpha
     \]
     \[
     y = 2mr \sin \left( \frac{\theta}{2} \right) \sin \alpha
     \]

6. **Analyze the locus of \( M \)**:
   - As \( B \) moves around the circle \( k \), \( \theta \) varies from \( 0 \) to \( 2\pi \).
   - The coordinates of \( M \) describe a circle centered at \( (r, 0) \) with radius \( 2mr \sin \left( \frac{\theta}{2} \right) \).

7. **Conclusion**:
   - The locus of point \( M \) is a circle centered at \( (r, 0) \) with radius \( 2mr \sin \left( \frac{\theta}{2} \right) \).

The final answer is the locus of point \( \boxed{ M } \) is a circle centered at \( (r, 0) \) with radius \( 2mr \sin \left( \frac{\theta}{2} \right) \).