In Exercises 21–40, eliminate the parameter t. Then use the re...

Question

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.

x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Write the corresponding rectangular equation for the following parametric equation by eliminating T. Draw a graph of the plane curve using the rectangular equation. Indicate the direction of the curve that is obtained by using arrows that correspond to the increasing values of T. Awesome. So it appears for this particular like particular problem we're asked to solve for three separate answers. So we're trying to figure out, firstly, we're trying to write a corresponding rectangular equation for this parametric equation, which the parametric equation is, as given to us as, X equals 5 + 3 multiplied by the cosine at T. And Y is equal to -2 + 4 multiplied by sin of T, and 0 is less than or equal to T and T is less than or equal to pi. So essentially we're trying to create a corresponding rectangular equation to follow this parametric equation by eliminating T. That's our first answer we're trying to solve for. Then we're asked to draw a graph of the plane curve using our rectangular equation that we solve for. And lastly, our last answers we're trying to indicate the direction of the curve that is obtained by using arrows that correspond to the increasing values of T. Awesome. So with that in mind, let's read off our multiple choice answers, noting that one answer is for our rectangular equation and the next answer is for the direction. Of the curve. So A is x minus 5 to 2 divided by 9 plus Y + 2 to the power of 2 divided by 16 is equal to 1 and the direction is right. B is X minus 52 divided by 9, plus Y + 22 divided by 16 is equal to 1 and left. C is x + 52 divided by 9 + Y minus 22 divided by 16 is equal to 1 and left, and finally d is X + 52 divided by 9 plus Y minus 22 divided by 16 is equal to 1 and right. Awesome. So first off, let us quickly write down our parametric equations again. So we have X is equal to 5 + 3 multiplied by the cosine of T, and we're also told that Y is equal to and let's write this down below, so we have a little bit more room. So we have Y is equal to -2 + 4 multiplied by sine of T. Fantastic. And our boundary is 0 is less than or equal to T and T is less than or equal to pi. Fantastic. So, in order to eliminate the parameter T, as we should recall and note, we need to start with the given parametric equations, which once again is X equals 5 + 3 multiplied by cosine a T, and Y equals -2 + 4 multiplied by sin a T. And we need to first solve for cosine of T and sine of T from these equations. So, from X equals, so from our X equals 5 + 3 multiplied by cosine of T, we will get that. So from this equation, we'll start with our X. So I'll make a little dot here. So with our X, we will get that cosine of T is equal to X minus 5 divided by 3, and from Y, we will get that sign of T, so we need to take sine of T. Is equal to. Y + 2 divided by 4. Fantastic. So now we need to use Piearus theorem identity, which states that, and let's write this in blue, write this in blue. So we need to recall and use the following identity that states that cosine of 2, so I said cosine, but I meant cosine squared. So this is cosine squared. Of T plus sine squared of T is equal to 1, and we need to substitute this in for the expression for cosine of T and sin of T. So when we do that, we will find that X minus 5, so X minus 5. Divided by 3 to the power of 2, plus Y + 2 divided by 4 to the power of 2 is all equal to 1. So we can simplify this expression to get our rectangular equation. So when we simplify this expression, we will simply get X minus 5 to 2 divided by 9 + Y + 2 divided by and don't forget that Y + 2 is squared, so Y + 22 divided by 16 is equal to 1, and that is our final answer for our rectangular equation. Hooray, we did it. So now we need to move on to our next few things that we're trying to solve for. So now we're trying to create a graph of our rectangular function, and then we need to figure out the direction of our curve. Awesome. So, with that said, I've gone ahead and plugged in our our A rectangular equation into my calculator. You can, I used a graphing calculator, but you can also use whatever graphing software that makes sense to you, whatever is easy to use. But essentially, it's really helpful if you create a table and then you use your table to plot your curve. So for our values of T, which is in our far left column, we have T equals 0, T equals pi divided by 2. And T equals pi. And then our next column is for X, so when the values of T equals 0, X is 8, when T is pi divided by 2, X is 5, and when T equals pi, X is 2, and similarly for Y, it's -2, 2, and 2. So when you plot these points on your graph, so when you see you have, and it's important to note that our standard plotting method is XY. So our value for X, so you follow the X axis, so when X equals 8, you go 8 along your ver so you go 8 along your horizontal X axis because you go, you go to the right 8 because it's a positive 8, and then you go down -2 because it's a negative, so that means you have to go down 2. So then you have your first point, which is 8.2. And then we have our value of X equals 5. So when X equals 5, so you go to the right 5, because it's positive, and you go up to B2. So that's our second point on our curve. And then lastly, when you go for X, so when like X equals 2, so we have 22, so you go to the right 2 because it's a positive 2, but then you have to go down to be with the negative symbol. Awesome. And then as we could see, our values for T increase going to the left. So that means, without a doubt, our final answer has to be multiple choice answer letter B, which once again is X minus 5 to 2 divided by 9 plus Y + 22 divided by 16 equals 1, and the direction of the curve is to the left. And that's it. We've officially solved for this problem. Hooray. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Parametric Equations

Eliminating the Parameter

Graphing and Orientation

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