Graph of x = sin2t, y = 2cost

To understand the graph represented by the equations x = sin2t, y = 2cost, we need to analyze each equation separately and then combine them to get the overall graph.

Equation x = sin2t

- The equation represents the x-coordinate of the point on the graph.
- The function sin2t oscillates between -1 and 1 as t varies.
- Therefore, the x-coordinate of the point also oscillates between -1 and 1 as t varies.
- This means that the graph of x = sin2t is a sinusoidal curve that oscillates between -1 and 1.

Equation y = 2cost

- The equation represents the y-coordinate of the point on the graph.
- The function cost oscillates between -1 and 1 as t varies.
- Therefore, the y-coordinate of the point also oscillates between -2 and 2 as t varies.
- This means that the graph of y = 2cost is a sinusoidal curve that oscillates between -2 and 2.

Combining the equations

- The two equations represent the x and y coordinates of the same point on the graph.
- Therefore, we can plot the points (x,y) on the graph where x = sin2t and y = 2cost for different values of t.
- As t varies, the point (x,y) traces out a curve on the graph.
- Since both x and y are sinusoidal functions, the curve traced out by the point (x,y) is also a sinusoidal curve.
- However, the curve is not a simple sinusoidal curve because the x-coordinate and y-coordinate have different amplitudes and periods.
- The curve traced out by the point (x,y) is a parabolic curve because it has a U-shape and is symmetric about the y-axis.
- Therefore, the graph represented by the equations x = sin2t, y = 2cost is a parabolic curve.

Conclusion

The correct answer is option 'B'. The graph represented by the equations x = sin2t, y = 2cost is a parabolic curve.