Identifying Parametric Equations in the Plane

Question

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

Accepted Answer

Identify the particle's path by finding a Cartesian equation for it. Graph this equation indicating both the direction of motion and the portion of the graph traced by the particle for the following parametric equations X equals 5 cosinet. Y equals 2 sint from t between 0 and 2 pi. We're also given a graph to plot our equation on. Now, to solve this, let's first solve for cosine and sine of t. Cosinet will end up being X divided by 5. Sint will end up being Y divided by 2. Now we can use our Pythagorean identity. Cosine squared t plus sine squared t equals 1. This will give us x divided by 52 plus Y divided by 22 equals 1. Or X 2 divided by 25 plus Y2 divided by 4 equals 1. This is the graph of an ellipse centered at the origin. Now from here We need to identify the portion traced. The firemen. The direction of motion And graph the equation. So We know this is going from 0 to 2 pi. This will end up being a full circuit of the ellipse. So, let's determine where it starts at. So we know then. That T equals 0. We have x equals 5 cosine is 0. And the Y equals to sinus 0, which gives us the starting point of 50. This means it starts on the rightmost part of our ellipse, which we will plot at 50. No. At pi divided by 2, we get X. Equals 0, Y equals 2. You have the 0.02. At T equals pi. We get x equals -5, Y equals 0 to give us -50. And finally, T equals 3 divided by 2 gives us X equals 0, Y equals -2. Give us a point of 0-2. Now, this is an ellipse, so it will start and end at the same place, meaning for T equals 2 pi, we end up getting our original point for T equals 0, 50. Now we can plot our points and graph our ellipse. So we mentioned before, the starting point is that. 50. And we graph 3 more points, 02. -50 and 0-2. Now we just draw a curve. For our ellipse. And we now know the direction, cause it's starting. At 50 and going counterclockwise. So. We can draw arrows to indicate the direction. This is the answer to our problem. OK, I hope to help you solve it. Thank you for watching. Goodbye.

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