A body floats in a liquid A of density `rho_(1)` with a part of it submerged inside liquid while in liquid B of density `rho_(2)` totally submerged inside liquid. The densities `rho_(1)` and `rho_(2)` are related as :

Options

A. `rho_(1)=rho_(2)`
B. `rho_(1)ltrho_(2)`
C. `rho_(1)gtrho_(2)`
D. nothing can be said

Answer

To solve the problem, we need to analyze the conditions under which a body floats in two different liquids, A and B, with given densities \(\rho_1\) and \(\rho_2\).

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- In liquid A (density \(\rho_1\)), the body is partially submerged, which means it floats.
- In liquid B (density \(\rho_2\)), the body is completely submerged.

2. **Applying Archimedes' Principle**:
- According to Archimedes' principle, a body will float if the weight of the liquid displaced by the submerged part of the body is equal to the weight of the body itself.
- For the body to float in liquid A, the density of the body must be less than the density of liquid A. Therefore, we can express this as:
\[ \text{Density of body} < \rho_1 \]

3. **Analyzing Liquid B**:
- In liquid B, since the body is completely submerged, it means that the buoyant force acting on the body is equal to the weight of the body. However, for the body to be fully submerged, the density of the body must be greater than the density of liquid B. Thus, we can express this as:
\[ \text{Density of body} > \rho_2 \]

4. **Combining the Inequalities**:
- From the two inequalities derived from the conditions in liquids A and B, we have:
\[ \rho_2 < \text{Density of body} < \rho_1 \]
- This implies that the density of the body lies between the densities of the two liquids.

5. **Establishing the Relationship**:
- Since the density of the body is greater than \(\rho_2\) and less than \(\rho_1\), we can conclude that:
\[ \rho_1 > \rho_2 \]

### Final Conclusion:
The relationship between the densities of the two liquids is:
\[ \rho_1 > \rho_2 \]