A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de-broglie wavelength of the particles varies cyclically between two values `lambda_(1), lambda_(2) with lambda_(1)gtlambda_(2)`. Which of the following statements are true?

Options

A. The particle could be moving in a circular orbit with origin as centre.
B. The particle could be moving in an elliptical orbit with origin as its focus.
C. When the de Broglie wavelength is `lamda_(1)`, the particle is nearer the origin than when its value is `lamda_(2)`.
D. Both (a) and (c )

Explanation

To solve the problem, we need to analyze the motion of a particle in a closed orbit around the origin, considering the variations in its de Broglie wavelength. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
The particle moves in a closed orbit due to a central force directed towards the origin. The de Broglie wavelength of the particle varies cyclically between two values, \( \lambda_1 \) and \( \lambda_2 \), with the condition that \( \lambda_1 > \lambda_2 \).

### Step 2: Relate de Broglie Wavelength to Velocity
According to the de Broglie hypothesis, the wavelength \( \lambda \) of a particle is given by:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 3: Set Up the Equations for the Two Wavelengths
From the information given:
- For \( \lambda_1 \):
\[ \lambda_1 = \frac{h}{m v_1} \quad \text{(1)} \]
- For \( \lambda_2 \):
\[ \lambda_2 = \frac{h}{m v_2} \quad \text{(2)} \]

### Step 4: Analyze the Relationship Between Wavelengths and Velocities
From equations (1) and (2), we can express the relationship between the wavelengths and velocities:
\[ \frac{\lambda_1}{\lambda_2} = \frac{v_2}{v_1} \]
Given that \( \lambda_1 > \lambda_2 \), it follows that:
\[ \frac{v_2}{v_1} < 1 \implies v_2 < v_1 \]
This indicates that the particle moves faster when it has a shorter wavelength.

### Step 5: Consider the Implications of the Particle's Motion
Since the particle is moving in a closed orbit and the force is directed towards the origin, the path of the particle is likely elliptical. The speeds \( v_1 \) and \( v_2 \) correspond to the particle’s speed at two different points in the orbit.

### Step 6: Determine the Shape of the Orbit
In an elliptical orbit, the speed of the particle is greater when it is closer to the focus (the origin in this case) and slower when it is farther away. Therefore, since \( v_2 < v_1 \), the particle is closer to the origin when it is at the point corresponding to \( \lambda_2 \).

### Conclusion
Based on the analysis, we conclude that the particle could indeed be moving in an elliptical orbit with the origin as one of its foci. Thus, the correct statement is that the particle could be moving in an elliptical orbit with the origin as its focus.

### Final Answer
The true statement is: The particle could be moving in an elliptical orbit with the origin as its focus.

---