A partical of mass M at rest decays into two Particles of masses
`m_1 and m_2` having non-zero velocities. The ratio of the de - Broglie wavelengths
of the particles `lambda_1| lambda_2` is
(a) `m_1//m_2` (b)`m_2// m_1` (c ) 1 (d)`sqrt,_2 // sqrt_1`

Explanation

To solve the problem, we need to find the ratio of the de Broglie wavelengths of two particles resulting from the decay of a particle at rest. Let's go through the solution step by step.

### Step 1: Understand the conservation of momentum
A particle of mass \( M \) is at rest and decays into two particles of masses \( m_1 \) and \( m_2 \) with velocities \( v_1 \) and \( v_2 \). According to the law of conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
Since the initial momentum is zero (the particle is at rest), we have:
\[ 0 = m_1 v_1 + m_2 v_2 \]
This implies:
\[ m_1 v_1 = -m_2 v_2 \]

### Step 2: Express the relationship between velocities
Taking the magnitudes, we can write:
\[ m_1 v_1 = m_2 v_2 \]
From this, we can express the ratio of the velocities:
\[ \frac{v_1}{v_2} = \frac{m_2}{m_1} \]

### Step 3: Use the de Broglie wavelength formula
The de Broglie 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 4: Write the expressions for the de Broglie wavelengths
For the two particles, we have:
\[ \lambda_1 = \frac{h}{m_1 v_1} \quad \text{and} \quad \lambda_2 = \frac{h}{m_2 v_2} \]

### Step 5: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio \( \frac{\lambda_1}{\lambda_2} \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{m_1 v_1}}{\frac{h}{m_2 v_2}} = \frac{m_2 v_2}{m_1 v_1} \]
Substituting the relationship we found earlier:
\[ \frac{\lambda_1}{\lambda_2} = \frac{m_2 v_2}{m_1 \left(\frac{m_2}{m_1} v_2\right)} = \frac{m_2}{m_1} \]

### Step 6: Conclusion
Thus, the ratio of the de Broglie wavelengths of the two particles is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{m_2}{m_1} \]
Therefore, the correct answer is option (b) \( \frac{m_2}{m_1} \).