if-a-and-b-are-fixed-numbers-find-parametric-equations-for-the-curve-that-consists-of-all-possible-2

Question

If $ a $ and $ b $ are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point $ P $ in the figure, using the angle $ 	heta $ as the parameter. The line segment $ AB $ is tangent to the larger circle.

Wen Zheng · Accepted Answer

The problem is, if a and B are fixed, the members find the parametric equations for the curve that consists of our impossible propositions of the pointy in the figure using the angle beta separator life segment it B is tangent to the larger circle. So you think you know it is a coordinate off? He is x y. Then we&#x27;re half axe is equal to which is equal to Oh, aye, um, steak and data, which is equal to a hams. Second data now why is he called, which is equal to all? See hams sign there, which is equal to eat House sign data. So the parametric equations for the curve. Our accidental toe ate ham sik in the data. Why&#x27;s he could beat hams, Sign data.

Wen Zheng · Answer

The problem is, if a and B are fixed the numbers by the Parametric equations for the proof that consists off our possible positions of the point p. The figure using the angle theta as a permitted then eliminates the perimeter. And I didn&#x27;t have any proof. So look at this graph here. O A. Is equal to B or B Z Goto, eh? And he is perpendicular to BP. So your half if the coordinate of p is x y on behalf of X is equal to o que, which is equal to okey hams Call sign, which is equal to a hams Co sign data. Why is the door to Q which is able tto away hams Sign dinner, which is what will be. Here&#x27;s This isn&#x27;t if this is a hams Oh, is it going to be here? This is being a anytime scientist. Then we have axe off a squire. Plus why over he squire, it&#x27;s equal to co sign data squire. Ah, sign data squire. When she is equal to what there&#x27;s a curve is my lips

Jeffrey Russell · Answer

we&#x27;re going to create a parameter ization for all possible points. P if a and B are fixed. So let&#x27;s find the X and Y in corn and of Pete in terms of A and B and CIA, where it stayed up will be our parameter. So first will find the X coordinate. So for the X coordinate, we want the length from the center of the graph here to right below P on the X axis. So we want the length from the center of the graph to this point. Well, if we consider this triangle extended down a little bit, this is a right triangle and so we can use data and a to find the length of this portion of the graph. So that&#x27;s X is equal to a Times Deco sign of Fada since we want to find the length of the adjacent side. Okay, now we&#x27;ll find the glamorization of why so for why we want to find this height in black here. Well, this high is the same as this height over here, since it&#x27;s a horizontal line right to the circle, and this forms another triangle where the length of this high pod news is be and we have the same angle, Sita. So this time you want to find the length of this side. So we will use be time sign of data to find the length of the opposite side. So this is the parameter ization here. Now we&#x27;re going to eliminate the parameter will solve for co signing, stayed on one side and sign of data for the other brander. And then we&#x27;ll add co sine squared data and science where data to eliminate the parameter. So we have co sign of data is equal to X over a and sign of data is equal to y over B. So then we have co sign squared data plus sign squared. Seita is equal of one. But by this these two equations here it is also equal to X over a square. Plus why over b squared. So here&#x27;s the equation. Once we&#x27;ve eliminated our parameter but notice this is the equation for an ellipse with one access being linked A and the other being length be the X axis being linked egg y axis being like Be so the curve traced out by P is this ellipse

Amrita Bhasin · Answer

okay for this. We know that we would use the Pythagorean theorem and we should end up with the square root of a squared times seeking It&#x27;s where data, The reason wise, Because we did the skirt of a squared plus a scored 10 scored data. And so now we&#x27;ve ended up with this, which we can further simplified to just be a Sikh in fate. And the reason why is because we&#x27;ve got a squared. But the whole thing scored, routed the squares a square It&#x27;s just cancel out. So a c can&#x27;t data. Therefore, we know, as you can tell from the graph that we would have why equals b signed data. So this is our first equation then going off with this from looking at the graph or second equation would be why equals b sine theta following the similar Potter Now to graft this, we know that we would have two circles. Then we would have a curve on the right and left inside. You can use a graphing instrument to graft this we would have won a circle like this and you know this circle would have a radius of two. So this point over here would be to over here, it would be too. And that negative too. In the negative, too. The outside circle would be two and 1/2. And then we know over here that we would have curves going like this, you can use like a graphing instrument. Edgar, off this we know this would be it. Approximately three run over here. In this case, it would be negative three.

Tom Greenwood · Answer

we&#x27;re given this picture of two concentric circles. Though Senator at the Oregon smaller circle has a radius of B, the larger circle has a radius of a and we&#x27;re making an angle. Say that here with the positive X axis and drawn the radius out through both circles. And then we have Point P, which is in line with a point on the small circle and land with a point on the larger circle. Now, if you remember from earlier discussion and homework problems, the Parametric equations for a circle Our X equals a times he co sign of the radius of Santa, and why equals a times a sign of data. So that would be the Parametric equation for my big circle. And I should have done my circles in different colors in my equations in different colors. Uh, for the smaller circle, we have X equals be times a call sign of data, and why would he will be times a sign of fate? Well, the first thing they asked us to do here was to find the parametric equations for the curve that he is going to trace out as we move our angle around the circle. So as we do, that pee is gonna move. And if we bring this angle down to along the X Axis P would be here on the outside circle. As we move it around, he&#x27;s gonna move in a path. And as we get to the y axis, he is actually gonna end up on the small circle and then continue around like that. So maybe you can imagine the path it&#x27;s gonna follow here as we continue and draw this. So I noticed that he is always lined up with the X coordinate of our bigger circle, and he is always lined up with the Y Coordinate, uh, are smaller circles. So there&#x27;s our parametric equation that we&#x27;re gonna have for the path of point p. We&#x27;re gonna have X equals a times a co sign of data and y equals b times a sign half there. So there&#x27;s the inter department of this problem, Part B. They asked us to graft the curve using a graphing calculator, Uh, with a equals three and B equals two. Well, that means our larger circle is gonna have a radius off. Three are smaller circles gonna have a radius off too. So as we plugged that in your calculator Three cool sign of data and to sign of failure when you put that in and graph it, you were going to see the point P is gonna trace. Like I said around like this and like so So it will have a distance of three size of three horizontally and the size of two going vertically on your graph. Okay, so there&#x27;s part. Be part. See them. They want us to eliminate the parameter and identify the curve. Well, we actually did something very similar to this back in problem number 36. Earlier, we dealt with an Ellipse and the Parametric equations. We kind of went the other direction. We started with the equation and turned it into the parameters. So here we have the Parametric versions. So if we divide both of these by A and we divide both sides here by B, that gives us co sign of Fada equals X over a sign of fatal equals x over beat. And like we did early in the problems, we can eliminate our parameter by using some trig identities. In this case, co sign squared off ADA plus sine squared of data cross one. And if co sign equals X over a co sign squared is gonna be X squared Over a squared and sign is why Overby so sine squared will be y squared over b squared And that only one and there is our Parametric.

Wen Zheng · Answer

the problem is that a finding the point of intersection after tangent lines to the curve Artie is they got to sign party time mighty cause I&#x27;m high tea at the points. Why? In tears ago to zero and tea if they go to zero point of off. So first before party pay I keep prom. Is they caught? You hide hams co sign high key high times Consign pikey on connective pie Time Sign high tea. So we have a zero from this. Is that you go to Hi. Hi. On zero. Are there a point foul from this is a call to zero zero. Make it your pie on our zero is equal. Chu zero zero one r they report five is equal to one one zero. Send equations off the timeline. Want t zero to zero. He is equal to zero. This is actually call Chiu. Hi, fi. Why go to he on sea? He could want when he is equal to zero point five times in the line is actually called one. Why? Because one on C go too negative pipes from this two equations with half the point of interception of the tank allies is Juan one war had to be illustrated by graphing the curve on both standing allies. Now look at is the grafts. Is it blue? One is a graph of the kerf on the yellow one and the green one as Tana lies so we can see intersection of this to analyze intersection point isthe one one one.

Problem

A curve, called a witch of Maria Agnesi, consists…

Problem 42 Easy Difficulty

If $ a $ and $ b $ are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point $ P $ in the figure, using the angle $ \theta $ as the parameter. The line segment $ AB $ is tangent to the larger circle.

Answer

Thus, the parametrization of the curve is $f(\theta)=a \sec \theta$ and $g(\theta)=b \sin \theta.$

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Thus, the parametrization of the curve is $f(\theta)=a \sec \theta$ and $g(\theta)=b \sin \theta.$

Calculus

Grace H.

Caleb E.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

James StewartCalculus

Grace H.

Caleb E.

Kristen K.

Joseph L.

James Stewart

Calculus