A particle moves in the XY-plane according to the law `x = kt, y = kt (1- alphat)`, where k and `alpha` are positive constants and t is time. The trajectory of the particle is

Options

A. `y = kx`
B. `y = x - (alpha x^(2))/(k)`
C. `y =- (ax^(2))/(k)`
D. `y = alpha x`

Explanation

To find the trajectory of the particle moving in the XY-plane according to the equations \( x = kt \) and \( y = kt(1 - \alpha t) \), we need to eliminate the time variable \( t \) and express \( y \) in terms of \( x \).

### Step-by-Step Solution:

1. **Start with the equations:**
\[ x = kt \]
\[ y = kt(1 - \alpha t) \]

2. **Solve for \( t \) from the equation for \( x \):**
\[ t = \frac{x}{k} \]

3. **Substitute \( t \) into the equation for \( y \):**
\[ y = k\left(\frac{x}{k}\right)\left(1 - \alpha\left(\frac{x}{k}\right)\right) \]

4. **Simplify the equation for \( y \):**
\[ y = x\left(1 - \frac{\alpha x}{k}\right) \]
\[ y = x - \frac{\alpha x^2}{k} \]

5. **Rearranging gives the final form of the trajectory:**
\[ y = x - \frac{\alpha}{k} x^2 \]

### Final Expression:
The trajectory of the particle is given by:
\[ y = x - \frac{\alpha}{k} x^2 \]