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A curve, called a witch of Maria Agnesi, consists of all possible positions of the point $ P $ in the figure. Show that the parametric equations for this curve can be written as $$ x = 2a\cot \theta \quad y = 2a \sin^2 \theta $$Sketch the curve.

$y-$ coordinate of $P$ is $y=2 a \sin ^{2} \theta$

03:18

Wen Z.

17:32

Bobby B.

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

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Boston College

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00:28

So in this problem we give them this diagram called the witch of Maria and Agnes E. And were asked to show that the parametric parametric equations for the curve which is the position of this point P. Out here. Okay. Our X equals to a co tangent data and Y equals to a sine squared data. Okay? So this is what we're trying to prove here. All right, so let us first notice on this diagram, a few other relationships here. I'm gonna call this point E up here where the line Y equals to a intersects the vertical axis. Okay, I noticed that this is a right angle here. And since this is data down here where this blue line through the point A. To see from the origin is an angle theta with the horizontal axis. Then this is also angle theta right there. I'm gonna call this point B. Right here where I went from the point A. All the way down to the X axis. And I'm going to call this point D. Where to go from Point B. All the way down to the horizontal axis here. Okay, I can also notice that this angle pence. This is a right triangle here. Or right angle here at the origin. This is Pi over 2- data, isn't it. Okay, now let's notice something here. Next the distance from C to D. Well that's just a straight line All the way up from the X axis. All the way up to this line Y equals to A. That means that distance is to A. And we can notice that the angle from see through the origin to D. His data we've enabled that one data down there that was given to us. So then we see that we have this big right triangle out here don't we? And so we can write that the tangent of data is what well tangent opposite over adjacent. So that's the distance from C. To D. Over The distance from 0 to D. Okay. Which means that the co tangent of theta is the distance from zero to D. Over the distance from C. To D. Then innit His co tangent is one over the tangent. All right. So this means we now have that The distance from 0 to D. Is the distance from C. To D. Times the co tangent of beta. But the distance from C. Diddy we just said over here was to A. So this is to a co tangent data and the distance from zero the D. Is the X coordinate. And so that one does hold we just proved it just showed it. Okay so next what can we see we can see that the distance from zero to A is Work from 0 to a. Okay that's this leg on the bottom of this right triangle isn't it? And so that's the distance from O. T. E times co sign of this angle right here and it By over 2- data. All right well that means that this is well the distance from oh e That's the whole vertical y distance in it. Which is to a right okay so this is to a and remember from our triggered a metric relationships that the co sign of pi over two minus theta is sign of data. Okay. So that means this is to a sign data. Now let's look at this triangle. Oh a B for a minute. Let's look at this triangle for a second. And we know that the sign of fada on a day to is sign 5th data right, is the opposite over their partners. So that means this is the distance from A. To B. Over. This is from 0 to a. So that gives us that the distance from O to A. I'm sorry I meant to do do it this way. The distance from A to B is the distance from O to a signed data. Yeah. And so therefore the distance from A to B is to a find data. Find data. Or in other words to a sine squared data. The distance from oh from A to B A B. That's the y coordinate for our point P. So we just so that formula come out, didn't we? And so therefore we have now shown that these are the parametric equations for the point P in this diagram

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