A particle moves in the x-y plane. It x and y coordinates vary with time t according to equations `x=t^(2)+2t` and `y=2t`. Possible shape of path followed by the particle is

Options

A. Straight line
B. Circle
C. Parabola
D. More information Is required to decide.

Answer

To determine the shape of the path followed by the particle moving in the x-y plane, we start with the equations given for the coordinates:

1. **Equations of Motion**:
- \( x = t^2 + 2t \)
- \( y = 2t \)

2. **Express \( t \) in terms of \( y \)**:
From the equation for \( y \):
\[ y = 2t \implies t = \frac{y}{2} \]

3. **Substitute \( t \) in the equation for \( x \)**:
Now, substitute \( t = \frac{y}{2} \) into the equation for \( x \):
\[ x = \left(\frac{y}{2}\right)^2 + 2\left(\frac{y}{2}\right) \]

4. **Simplify the equation**:
\[ x = \frac{y^2}{4} + y \]

5. **Rearranging the equation**:
To express this in a standard form, we can rearrange it:
\[ x = \frac{y^2}{4} + y \implies 4x = y^2 + 4y \]

6. **Complete the square**:
We can complete the square for the \( y \) terms:
\[ 4x = (y^2 + 4y + 4) - 4 \implies 4x + 4 = (y + 2)^2 \]
Thus, we have:
\[ (y + 2)^2 = 4(x + 1) \]

7. **Identify the shape**:
The equation \( (y + 2)^2 = 4(x + 1) \) is in the standard form of a parabola, which opens to the right. The vertex of this parabola is at the point \( (-1, -2) \).

### Conclusion:
The possible shape of the path followed by the particle is a **parabola**.