The coordinates of a particle moving in XY-plane very with time as `x=4t^(2),y=2t`. The locus of the particle is

Options

A. Statement-I is true, Statement-II is true, Statement-II is correct explanation for statement-I
B. Statement-I is true, Statement-II is true, Statement-II is NOT a correct explanation for statement-I
C. Statement-I is true, Statement-II is false.
D. Statement-I is false and Statement-II is true.

Answer

To find the locus of the particle whose coordinates vary with time as \( x = 4t^2 \) and \( y = 2t \), we will eliminate the parameter \( t \) from these equations. Here’s a step-by-step solution:

### Step 1: Write down the equations
We start with the given equations:
\[ x = 4t^2 \]
\[ y = 2t \]

### Step 2: Solve for \( t \) in terms of \( y \)
From the equation for \( y \), we can express \( t \) in terms of \( y \):
\[ y = 2t \implies t = \frac{y}{2} \]

### Step 3: Substitute \( t \) into the equation for \( x \)
Now, we substitute \( t = \frac{y}{2} \) into the equation for \( x \):
\[ x = 4t^2 = 4\left(\frac{y}{2}\right)^2 \]

### Step 4: Simplify the equation
Now, simplify the expression:
\[ x = 4 \cdot \frac{y^2}{4} = y^2 \]

### Step 5: Rearrange the equation
We can rearrange this equation to express it in a standard form:
\[ y^2 = x \]

### Conclusion
The locus of the particle is described by the equation \( y^2 = x \), which represents a parabola that opens to the right.