Finding Parametric Equations

Find parametric ...

Question

Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Accepted Answer

Hello. In this video, we are going to be finding the parametric equations and a parameter interval for the motion of a particle that starts at 0. B and traces the ellipse X2 divided by A2 plus Y2 divided by B2 equal to 1, exactly once counterclockwise. Ensuring that the x is expressed in terms of sin of t, y is expressed in terms of cosinet, and the parameter interval begins at t is equal to 0. So, let's go ahead and start this problem by identifying the interval of the parameter. So, the problem tells us that for the given ellipse, we want to rotate around the ellipse exactly once counterclockwise. So if we think about a rotation counterclockwise, counterclockwise means that we are traveling in the positive direction with respect to a unit circle. Now, because we want to travel exactly once, we want to make one full rotation in the counterclockwise direction. Now, a full rotation is going to be from the angle 0 to the angle 2 pi. So if we name a parameter T, then the interval that we want is for T to be between the values of 0 and 2 pi. So this is going to be the interval for the for the parameter that we are going to solve for. Now let's take a look at the given equation. Now here we are given the general equation of an ellipse, that is X2 divided by A2 plus Y2 divided by B2 is equal to 1. And we know that the starting point of this particle is going to be at 0.B. Now here, if we want to rotate about the. If we want to rotate about the ellipse exactly once counterclockwise, then we want to find values of X and Y that satisfy the parametric equation, sin 2 T plus cosine 2 T equal to 1. In order for this to happen, we need to define x as the parametric equation a multiplied by sin of t. And we are going to define Y as B multiplied by sin of T. So these are going to be the general forms of our parametric equation, but now we need to find the specific values that satisfy the starting point at 0. B. So, let's go ahead and find where to start at 0.B when a parameter T is equal to 0. Now, if we take a look at our parametric equation for the A, we want the output of a sin of T to equal to 0, because the X component of this point is 0. Well, there's only one value oft that is going to satisfy this equation, and that is going to be when t is equal to 0. By doing a quick check, if we allow A toe or x equal to a multiplied by sin of zero, then the output is going to be 0. So that is going to be the value of x. Now if we take a look at y, we know that y is equal to b multiplied by cosine of zero or cosine of t. Now in this case here, if we plug in t is equal to 0, then we are left with y is equal to B multiplied by 1, which is going to be B. Now we want the y component to be lowercase b. That means that in this case here, we're going to allow B to equal to lowercase b. So, with that being said, the next thing we're going to do is we're going to define the direction. Now here for direction. We want x equal to a sin of t to move counterclockwise. Now, in order for this to move counterclockwise, we need to define this as a negative value, so X must equal to -a multiplied by sin of T. And so with that being said, this is going to be our equation for the value of x, and for the value of y, we are going to have y is equal to lowercase b multiplied by cosine of t. And as we stated earlier, the interval of the parameter is going to be between 0 and 2 pi. So this is going to be the solution to the problem, with the overall solution being option C. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Finding Parametric EquationsFind parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².a. once clockwise.(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

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Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)