Assertion If velocity of charged particle in a uniform magnetic field at some instant is `(a_(1)hat(i)-a_(2)hat(j))` and at some other instant is `(b_(1)hat(i)+b_(2)hat(j))`, then
`a_(1)^(2)+a_(2)^(2)=b_(1)^(2)+b_(2)^(2)`
Reason Magnetic force cannot change velocity of a charged particle.

Options

A. If both Assertion and Reason are true and Reason is the correct explanation of Assertion.
B. If both Assertion and Reason are true but Reason is not correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If Assertion is false but Reason is true.

Answer

To solve the question, we need to analyze the assertion and the reason provided.

### Step 1: Understanding the Assertion
The assertion states that if the velocity of a charged particle in a uniform magnetic field at one instant is given by the vector \((a_1 \hat{i} - a_2 \hat{j})\) and at another instant by \((b_1 \hat{i} + b_2 \hat{j})\), then the magnitudes of these velocities are equal.

### Step 2: Kinetic Energy Analysis
The kinetic energy \(K\) of a charged particle is given by the formula:
\[ K = \frac{1}{2} m v^2 \]
where \(v\) is the magnitude of the velocity vector.

For the first velocity vector:
\[ v_1 = \sqrt{a_1^2 + a_2^2} \]
Thus, the kinetic energy at the first instant is:
\[ K_1 = \frac{1}{2} m (a_1^2 + a_2^2) \]

For the second velocity vector:
\[ v_2 = \sqrt{b_1^2 + b_2^2} \]
Thus, the kinetic energy at the second instant is:
\[ K_2 = \frac{1}{2} m (b_1^2 + b_2^2) \]

### Step 3: Equating Kinetic Energies
Since the magnetic force does no work on the charged particle, the kinetic energy remains constant:
\[ K_1 = K_2 \]
This leads to:
\[ \frac{1}{2} m (a_1^2 + a_2^2) = \frac{1}{2} m (b_1^2 + b_2^2) \]
Canceling \(\frac{1}{2} m\) from both sides, we get:
\[ a_1^2 + a_2^2 = b_1^2 + b_2^2 \]
This confirms the assertion is true.

### Step 4: Analyzing the Reason
The reason states that the magnetic force cannot change the velocity of a charged particle. This statement is partially true; while the magnetic force does not change the speed (magnitude) of the particle, it can change the direction of the velocity vector.

### Conclusion
The assertion is true because the magnitudes of the velocities at two different instants are equal due to the conservation of kinetic energy in a magnetic field. However, the reason is false because the magnetic force does change the direction of the velocity, even though it does not change the speed.

### Final Answer
- Assertion: True
- Reason: False