The position coordinates of a particle moving in the xy plane are given as x = 2t^2 and y = 4t, where t represents time.

To determine the locus of the particle, we need to eliminate the variable t and express the relationship between x and y.

Let's start by eliminating t from the given equations:

From equation (1): x = 2t^2
Rearranging the equation, we get: t^2 = x/2

From equation (2): y = 4t
Rearranging the equation, we get: t = y/4

Substituting the value of t from equation (2) into equation (1), we have:

(y/4)^2 = x/2
Simplifying the equation, we get: y^2/16 = x/2
Multiplying both sides of the equation by 16, we have: y^2 = 8x

Now, we have the relationship between x and y. By analyzing this equation, we can determine the shape of the locus.

Analyzing the equation y^2 = 8x, we can identify the shape:

1. The coefficient of x is positive, indicating that the locus is not a hyperbola (which has a difference of squares).
2. The equation is in the form y^2 = 4ax, which represents a parabola.

Therefore, the locus of the particle is a parabola.

Explanation:

- The position coordinates of the particle are given by x = 2t^2 and y = 4t, representing the particle's position in the xy plane at different times.
- By eliminating the variable t from the given equations, we arrive at the equation y^2 = 8x.
- Analyzing this equation, we can determine that the locus of the particle is a parabola.
- The coefficient of x is positive, indicating that the parabola opens to the right.
- The equation is in the form y^2 = 4ax, which is a standard form of a parabola.
- The value of a is 2, indicating that the focus of the parabola is at (a, 0) = (2, 0) and the directrix is x = -a = -2.
- The parabola will be symmetric with respect to the y-axis.
- As time increases, the particle moves along the parabola, with its x-coordinate increasing quadratically and its y-coordinate increasing linearly.
- The shape of the parabola can be visualized as a curve opening to the right, with the vertex at the origin (0, 0).

Therefore, the correct answer is c) A parabola.

It is a parabola.x=2t^2.so y^2=8x. which is of the form y^2=4ax