The co-ordinates of a particle moving in xy-plane vary with time as `x="at"^(2),y="bt"`. The locus of the particle is :

Options

A. Parabola
B. Circle
C. Straight line
D. Ellipse

Answer

To find the locus of the particle whose coordinates in the xy-plane vary with time as \( x = at^2 \) and \( y = bt \), we need to eliminate the parameter \( t \) and find a relationship between \( x \) and \( y \).

### Step-by-step Solution:

1. **Write down the equations for x and y:**
\[ x = at^2 \]
\[ y = bt \]

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

3. **Substitute \( t \) into the equation for \( x \):**
Substitute \( t = \frac{y}{b} \) into the equation for \( x \):
\[ x = a\left(\frac{y}{b}\right)^2 \]

4. **Simplify the equation:**
\[ x = a \cdot \frac{y^2}{b^2} \]
This can be rewritten as:
\[ x = \frac{a}{b^2} y^2 \]

5. **Identify the type of curve:**
The equation \( x = k y^2 \) (where \( k = \frac{a}{b^2} \)) represents a parabola that opens to the right.

### Final Result:
The locus of the particle is given by the equation:
\[ x = \frac{a}{b^2} y^2 \]
This is the equation of a parabola.