An iron rod of length 2l is sliding on two mutually perpendicular lines . Find the locus of the midpoint of the rod.

Answer

To find the locus of the midpoint of an iron rod of length 2L sliding on two mutually perpendicular lines, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Coordinate System**:
Let the two mutually perpendicular lines be the X-axis and Y-axis. The origin (0,0) is where these two axes intersect.

2. **Position the Rod**:
Consider the rod of length 2L sliding in such a way that its endpoints are on the axes. Let the endpoints of the rod be at points (0, A) on the Y-axis and (B, 0) on the X-axis.

3. **Identify the Midpoint**:
The midpoint M of the rod can be represented as M(X, Y). The coordinates of the endpoints of the rod are (0, A) and (B, 0).

4. **Use the Midpoint Formula**:
The coordinates of the midpoint M can be calculated using the midpoint formula:
\[ M(X, Y) = \left(\frac{0 + B}{2}, \frac{A + 0}{2}\right) = \left(\frac{B}{2}, \frac{A}{2}\right) \]
Therefore, we have:
\[ X = \frac{B}{2} \quad \text{and} \quad Y = \frac{A}{2} \]

5. **Relate the Length of the Rod**:
The length of the rod is given as 2L. The distance between the endpoints (0, A) and (B, 0) can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(B - 0)^2 + (0 - A)^2} = \sqrt{B^2 + A^2} \]
Setting this equal to the length of the rod, we have:
\[ \sqrt{B^2 + A^2} = 2L \]

6. **Square Both Sides**:
Squaring both sides gives:
\[ B^2 + A^2 = (2L)^2 = 4L^2 \]

7. **Substitute for B and A**:
Since we have \( B = 2X \) and \( A = 2Y \), substituting these into the equation gives:
\[ (2X)^2 + (2Y)^2 = 4L^2 \]
Simplifying this, we find:
\[ 4X^2 + 4Y^2 = 4L^2 \]

8. **Divide by 4**:
Dividing the entire equation by 4 results in:
\[ X^2 + Y^2 = L^2 \]

### Conclusion:
The locus of the midpoint of the rod is a circle with radius L centered at the origin (0,0).