During an experiment, an ideal gas is found to obey a condition `(p^2)/(rho) = "constant"`. (`rho` = density of the gas). The gas is initially at temperature (T), pressure (p) and density `rho`. The gas expands such that density changes to `rho//2`.

Options

A. The pressure of the gas changes to `sqrt(2)P`.
B. The temperature of the gas changes to `sqrt(2)T`.
C. The graph of the above process on the `P-T` diagram is parabola.
D. The graph of the above process on the `P-T` diagram is hyperbola.

Explanation

To solve the problem, we need to analyze the relationship given by the condition \(\frac{p^2}{\rho} = \text{constant}\) and how it affects the pressure, temperature, and the graphical representation in the \(P-T\) diagram when the gas expands and its density changes from \(\rho\) to \(\frac{\rho}{2}\).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
The condition states that \(\frac{p^2}{\rho} = \text{constant}\). This implies that \(p^2\) is directly proportional to \(\rho\):
\[ p^2 \propto \rho \quad \Rightarrow \quad p \propto \sqrt{\rho} \]

2. **Initial and Final Conditions**:
- Initial density: \(\rho_{\text{initial}} = \rho\)
- Final density: \(\rho_{\text{final}} = \frac{\rho}{2}\)

3. **Relating Initial and Final Pressures**:
From the proportionality derived earlier, we can write:
\[ \frac{p_{\text{initial}}}{p_{\text{final}}} = \sqrt{\frac{\rho_{\text{initial}}}{\rho_{\text{final}}}} \]
Substituting the values:
\[ \frac{p}{p_{\text{final}}} = \sqrt{\frac{\rho}{\frac{\rho}{2}}} = \sqrt{2} \]
Rearranging gives:
\[ p_{\text{final}} = \frac{p}{\sqrt{2}} = \frac{p}{\sqrt{2}} \quad \Rightarrow \quad p_{\text{final}} = \frac{p}{\sqrt{2}} \]

4. **Checking the First Option**:
The first option states that the pressure changes to \(\sqrt{2}p\). Since we found that \(p_{\text{final}} = \frac{p}{\sqrt{2}}\), this option is incorrect.

5. **Using the Ideal Gas Law**:
The ideal gas law states:
\[ \rho = \frac{pM}{RT} \]
Rearranging gives:
\[ p = \frac{\rho RT}{M} \]

6. **Finding the Relationship Between Pressure and Temperature**:
Since we know that \(p^2 \propto \rho\), we can also express it in terms of temperature:
\[ p^2 \propto \frac{pM}{RT} \quad \Rightarrow \quad p \propto \frac{1}{T} \]
Therefore, we can write:
\[ pT = \text{constant} \]

7. **Relating Initial and Final Temperatures**:
Using the relationship \(p_{\text{initial}}T_{\text{initial}} = p_{\text{final}}T_{\text{final}}\):
\[ pT = \frac{p}{\sqrt{2}}T_{\text{final}} \quad \Rightarrow \quad T_{\text{final}} = \sqrt{2}T \]

8. **Checking the Second Option**:
The second option states that the temperature changes to \(\sqrt{2}T\). Since we derived \(T_{\text{final}} = \sqrt{2}T\), this option is correct.

9. **Analyzing the Graph on the \(P-T\) Diagram**:
From the relationship \(p \propto \frac{1}{T}\), we see that pressure is inversely proportional to temperature. This indicates that the graph of pressure versus temperature will be a hyperbola.

10. **Checking the Third and Fourth Options**:
- The third option states that the graph is a parabola, which is incorrect.
- The fourth option states that the graph is a hyperbola, which is correct.

### Final Answers:
- The correct options are:
- Second option: Temperature changes to \(\sqrt{2}T\) (Correct)
- Fourth option: The graph of the above process on the \(P-T\) diagram is a hyperbola (Correct)