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 = 5 sec t, y = 3 tan t

x = 5 sec t, y = 3 tan t

Accepted Answer

Hello 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 problem we're asked to solve for three separate things. Firstly, we're trying to figure out the corresponding rectangular equation for the following parametric equation, which is given to us as by the problem itself as. X equals 4 multiplied by cosicant of T, and Y equals 2 multiplied by cotangent of T. Awesome. So with that in mind, we're 1st, 1st off, our first goal is we're trying to create a corresponding or rectangular equation to the following para parametric equation. That's our first step. Our first answer we're trying to solve for. Our second answer that we're trying to solve for is we're trying to create a graph of our new rectangular equation that we solve for. That's our second answer. And our third and final answer is we're trying to indicate the direction of the curve. That is obtained by using arrows that correspond to the increasing values of T. So we're going to be drawing arrows on our curve to show the increasing value of T. So it's either going to be in the downwards direction or upwards direction, vice versa. So with that said, let's read off our multiple choice answers to see what our final answer pair might be, and note that we have an answer for the rectangular equation and then an answer for the direction of the curve. So A is X2 divided by 4 minus X2 divided by 16 is equal to 1, and down. B is X2 divided by 4 minus y2 divided by 16 equals 1 and up C is X2 divided by 16 minus y2 divided by 4 equals 1 and up, and D is X2 divided by 16 minus y2 divided by 4 equals 1 and down. OK. So first off, as we should recall and note, in order to eliminate the parameter T, we need to recall and use trigon like trig identities for cosicant of T and cotangent of T. So we need to recall a note that Cosicant squared of T minus cotangents, so cotangent squared of T is equal to 1. So once again cosicant squared of T minus cotangent squared of T is equal to 1 recalling and using our trig identities and from X equals 4 multiplied by cosicant of T, we will get that coicant, so cosicant. Of T is equal to X divided by 4. Fantastic. And our next step now is we need to note that for Y equals 2 multiplied by cotangent of T, so cotangent. Of tea We will get that O tangent. Of T is equal toy divided by 2. So when we square both of these equations and subtract them, we will get X2 divided by 16 minus Y2 divided by 4, and then we're going to say that this is all equal to 1. And as we could see, as we should be able to recall and recognize that this is the equation for a hyperbola with a center at 00, so it's centered at the origin. So, our next step now is we need to recall and note that there is a semi-transverse axis at A equals 4, so let's make a note of that. So A equals 4. Along the X axis. Awesome. And we're, it's important to note that we also have a semi-conjugate axis at B equals 2 along the Y axis. Fantastic. And then we have to also recall and note that the asymptote of our hyperbola is we need to recall a note that and we'll say, we'll write our asymptotes in blue. So we have Y equals plus or minus B divided by A. And we also have, and this is the formatting for asymptotes for a hyperbola, and we say that X is equal to plus or minus 12 multiplied by X. So, Let us quickly note now that as, and let's make a note that as T approaches 0 from the right. We can state that. Cosicant of T, so cosicant of T will approach infinity. So, with that said, we can go ahead and say that as X approaches infinity. We can say Y also approaches infinity. And as he approaches i from the left, so whenever you see a positive sign in the power, that means it's from the right, and if you see a negative symbol in the power, that means from the left. So as T approaches i from the left, Posyant of tea is going to approach infinity. And cotangent of T will approach negative infinity. So cotangentity will approach negative inf infinity. So, therefore, as a result, X will approach infinity. And why will approach negative infinity. Awesome. So now we can officially draw a graph of our rectangular function, which our final answer for the rectangular function was X2 divided by 16 minus Y2 divided by 4 equals 1. So when we graph our rectangular function, we will get the following graph. So as we could see, When we plot our graph here, we have two hyperbolas, which are shaped like a butterfly. I joke and I call these butterfly graphs because you could see the wings of our hyperbola or like the wings of a butterfly. And as you could see, since following our boundaries, so let's go look. Get our boundaries here. For as T approaches 0 from the right and as T approaches i from the left, you could see that the values of T increase going downwards. So we have arrows pointing downwards, so going down or curve for both sides of our hyperbola. As you can see. And then we have our diagonal lines, uh, the diagonal dotted line in green for our one asymptote, and then we have our dotted purple line for our other asymptote, and these are for our X and Y asymptotes. And then as you could see, we have it. Uh, our asymptotes cross through the origin point, which is at 0.0. So I'll make a note of that. So our origin is at 0.0, as you could see at the center of our graph. And it's very important to note that these curves, which are curves are in red, these curves will get super, super close to these asymptoes, but they will never touch. They will never cross or intersect these asymptotes. So they'll just get closer and closer and closer to these asymptotees going up and up and up and forever for infinity. And that's it. We've officially solved for this problem. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter D, which once again is X2 divided by 16 minus Y2 divided by 4 is equal to 1, and the direction is down, so downwards, as you can see, because the increasing values of T are going down the curve. And that's it. 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 and Parameter Elimination

Trigonometric Identities

Curve Sketching and Orientation

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