Assertion: If the positon vector of a particle moving in space is given by `vecr=2thati-4t^(2)hatj`, then the particle moves along a parabolic trajector.
Reason: `vecr=xhati+yhatj` and `vecr=2thati-4t^(2)j`
`rArr y=-x^(2)`.

Options

A. If both assertion `&` Reason are True `&` the Reason is a corrrect explanation of the Asserion.
B. If both Assertion `&` Reason are True but Reason is not correct explanation of the Assertion.
C. If Assertion is Trie but the Reason is False.
D. If both Assertion `&` Reason are false

Answer

To solve the problem, we need to analyze the assertion and the reason provided in the question step by step.

### Step 1: Understand the Position Vector
The position vector of the particle is given as:
\[ \vec{r} = 2t \hat{i} - 4t^2 \hat{j} \]
This means that the x-component of the position vector is \( x = 2t \) and the y-component is \( y = -4t^2 \).

### Step 2: Express Time in Terms of x
From the equation \( x = 2t \), we can express time \( t \) in terms of \( x \):
\[ t = \frac{x}{2} \]

### Step 3: Substitute t into the y-component
Now, we substitute \( t = \frac{x}{2} \) into the y-component \( y = -4t^2 \):
\[ y = -4\left(\frac{x}{2}\right)^2 \]
Calculating this gives:
\[ y = -4 \cdot \frac{x^2}{4} = -x^2 \]

### Step 4: Identify the Type of Trajectory
The equation \( y = -x^2 \) represents a parabola that opens downwards. This confirms that the particle moves along a parabolic trajectory.

### Step 5: Conclusion
Both the assertion and reason are true. The assertion correctly states that the particle moves along a parabolic trajectory, and the reason provides the correct mathematical relationship that leads to this conclusion.

### Final Answer
Both the assertion and reason are true, and the reason correctly explains the assertion.

---