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 = 1 + 3 cos t, y = 2 + 3 sin t; 0 ≤ t < 2π

Accepted Answer

Welcome back, everyone. In this problem, we want to write the corresponding rectangular equation for the parametric equation X equals four plus seven, multiplied by the cosine of ty equals six plus 11, multiplied by the sign of T and T is between zero and two pi. And we want to eliminate T for our, for our to get our rectangular equation. After we do that, we're going to draw a graph of the plane curve using the rectangular equation and then indicate the direction of the curve obtained corresponding to increasing values of T using arrows on our graph that we have both X and Y are bounded by negative 2020 on their axes respectively. Now, what, what are we doing here? What do we already know it? I mean, it looks like a lot. But what it's essentially asking us to do is to get our parametric equation to become a regular equation. A rectangular equation by eliminating T T is a common parameter that relates X and Y. So essentially, we're trying to just get an equation that has only XS and Ys. We know that X is a function of the cosine and Y is a function of sine. And when we think about it, that parametric curve usually, or typically looks like an ellipse. So we know that we're going to want to plot something that looks like that on the X and Y axis, we also know that since we're dealing with the cosine and sine function, they're related by the Pythagorean identity because the square of the cosine and the square of the sign when we add both of those together, they equal one. So if we are going to get that rectangular equation first, we may need to solve for the value of the cosine of T and the sign of T in order for us to put it into our Pythagorean identity, because that essentially will eliminate T. And once we do that, then we will be able to go ahead and draw our graph. So let's start there. So I recall, well, even before we apply the identity, let's solve for the sign and the co sign. OK. So let me put the cosine in red, OK. So given X equals four plus seven multiplied by the cosine of T. That means the cosine of T would be equal to X minus four or rather seven multiplied by the cosine of T equals X minus four. And then the core set of T would be X minus four all divided by seven. Next, let me put my equation for the sine function in blue. So given Y equals six plus 11 multiplied by the sign of T, then 11 multiplied by the sign of T equals Y minus six. And the sign of T would be Y minus six all divided by 11. No, we have the cosine of T and the sign of T expressed in terms of X and Y. So we can use our py, so let's do that. So I'm gonna put that in black. So using the Pythagorean identity, OK. And we know the identity says that the square of the cosine of T plus the square of the sign of T equals one, then substituting the values for the cosine. And the sign that means X minus four divided by seven all squared. OK plus Y minus six divided by 11 or squared. Oops I should have, let me put that in blue just to show the difference between both equations out of that square equals one. Now I can start to expand. So X minus four are squared divided by seven squared, which is 4 to 9. And then we're gonna have Y minus six R squared divided by 11 squared, which is 121 and all of that equals one. So no, we have a formula, a rectangular equation. That is, that doesn't have any parametric values in there. We have eliminated T essentially OK. So that's good. So we've done the first part to it. The next part of our question wants us to draw a graph of the plane curve using that rectangular equation. OK. So if we're going to draw that para that, that graph, we know that it looks like a, an ellipse. And for that ellipse, that means essentially we need probably four points to plot. So we have a range of points that we could choose, we could choose them at ransom at random. Don't take the points for ransom at random. But let's let's be smart about this. OK? If we need those points, we want to choose some points that makes it easy to calculate our values for X and Y. So I'm going to draw a little table here for the values of X and Y. OK? And we, as we said, we want about four values and we need some values that are easy to work with. So let's think about it here. I have X minus four R squared divided by 49. If I choose a value of X, I'm going to want a value that's going to simplify this as easy as possible. For example, I'd want a value of X that would simplify to one if this fraction is going to be simplified to one, that means the numerator X minus four or squared would have to be equal to 4 to 9 as well. And that means X minus four would have to be equal to seven. That would be the best approach. So what can I, what number when I subtract four it makes it either seven or negative seven. Well, I could use value of negative three for X because negative three minus four makes it negative seven. Or I could also let X be equal to 11. So no, I already have two easy numbers that I have to work with. How about making that number also equal to zero? Because if it equals zero, then I can easily eliminate it and just focus on Y minus six, all squared, all divided by 100 and 21 equals one if I can eliminate that variable. So what can I put in the bracket in the parentheses of X minus four to make it zero? Well, I could let X equal four. So I have three possible values and I'm going to work them. Hopefully I'll be able to use one to get two possible values of Y. So let me start with X being equal to negative three. When X equals negative three, then my equation would be negative three minus four, all squared divided by 49 plus Y minus six, all squared divided by 100 and 21. And that equals one. I know the reason I chose this value is because all of this is going to be negative seven squared, which is 49 and 49 divided by 49 equals one. And that means my equation will become one plus Y minus six squared divided by 100 and equals one and no, if I cancel one from both sides, that means Y minus six all squared divided by equals zero. And if that's the case, that means Y minus six R squared would be equal to zero, which means Y minus six equals zero. And that tells us Y equals six. So now we get a nice easy value for yy equals six. And that's one coin already negative 36. Let's try it out when X equals 11, right? So I'm going to for my space here, let me just replace my values of negative three with 11. OK? Because guess what? I already know what it's going to work out to be, that's going to be seven squared divided by 49 which would equal one and no, that would be Y minus X squared equals zero, Y minus six equals zero and Y equals six, even when I change my value for 11, I would still get Y being equal to six. Now, finally, what if we let X be equal to four? Well, let's see. Let me come down here a bit. So when we let X equal four, then our equation is going to become four minus four R squared divided by 49 plus Y minus six R squared, divided by 121 equals zero. No, if that's the case four minus four, again, I chose it to make this be equal to zero. And that tells us then that Y minus six all squared divided by equals one. OK? Because remember that was what the equation was not zero. It should have been one. My apologies to know that means why these were canceled, these wouldn't exist. And Y minus six R squared equals 121. If I go ahead, then that means Y minus six equals the square root. Cause if I square root both sides Y minus six equals positive or negative 11 plus or minus 11. And that means that Y would therefore be equal to positive one minus 11 plus six. So those are two possible values because when 11 is positive, Y equals 11 plus six, which is 17. And when 11 is negative, Y equals, let me actually write it out because that means Y equals positive 11 plus six, which is 17 and it would equal negative 11 plus six, which is negative five. So therefore, when X equals four, Y could be equal to negative five or Y could also be equal to 17. So that's two possible values. So our coordinates could be negative 36, 11 64, negative five and 4 17. So remember for four, there are two possible values know that we have the plans, we can go ahead and plot our graph. So let's do that. So I'm gonna carry these points with me. OK? Let me copy them and let me go back up to my graph and I'm gonna put these points, especially on the left, I should even draw like a, a square around them just for everybody to know what we're dealing with. So I highlight them dear on the right, know that we have our points, we can go ahead and plot. So when X is negative three, Y is six, so negative three and six, which would be right there when X is 11, Y is six. So 11 and six would be right here when X is four, Y is negative five and when X is four, Y is also 17. OK? Know that we have our four points, we can go ahead and draw our ellipse, which would probably look something like this. Please don't judge my ellipse. I try to, OK, let me draw it one more time just to get it looking a little better, a little curve here when you're drawing it though. One thing that helps is your French curve. So it shouldn't have that little dent in the side. But you, I think you basically get the idea. It's really hard to draw that curve on a, on a tablet. But finally, that would be our rectangular equation. And our parametric equation for the graph of X equals four plus seven multiplied by the cosine of ty equals six plus 11, multiplied by the sine of T and T is between zero and two pi. But this was one of those longer videos thank you so much for watching. I really hope that this video helped and I know that you did great, see you next time.

Key Concepts

Parametric Equations

Eliminating the Parameter

Orientation and Sketching of Parametric Curves

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