Introduction:
In structural analysis, the conjugate beam method is a technique used to determine the deflection and slope of a real beam. It involves converting the real beam into a hypothetical beam called the conjugate beam, which has the same length as the real beam but with certain modifications. One of the modifications is the support conditions at the ends of the beams.

Explanation:
The free end of the conjugate beam corresponds to a fixed support in the real beam. This means that the real beam has a fixed support at that particular end. Let's understand why this is the case.

Conjugate Beam Method:
To analyze the deflection and slope of a real beam using the conjugate beam method, we convert the real beam into a conjugate beam by applying certain rules:

1. Length and Load:
- The length of the conjugate beam is the same as the real beam.
- The loads on the conjugate beam are equal in magnitude to the moments at the corresponding locations on the real beam.

2. Support Conditions:
- The support conditions on the conjugate beam are related to the types of supports present in the real beam.

Fixed Support:
A fixed support in a real beam provides both vertical and horizontal constraints to the beam. This means that the beam cannot rotate or translate at that particular support. In the conjugate beam, the fixed support is replaced by a free end.

Corresponding Support:
The reason the free end of the conjugate beam corresponds to a fixed support in the real beam is that the fixed support constrains the rotation and translation of the beam. In the conjugate beam, the free end allows for rotation and translation, mimicking the behavior of a fixed support in the real beam.

Conclusion:
In the conjugate beam method, the free end of the conjugate beam corresponds to a fixed support in the real beam. This is because a fixed support in the real beam constrains the rotation and translation of the beam, which is represented by the free end in the conjugate beam. Understanding the correspondence between the support conditions in the real beam and the conjugate beam is crucial for accurate analysis of beam deflections and slopes.