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The figure shows a fixed circle $ C_1 $ with equation $ (x - 1)^2 + y^2 = 1 $ and a shrinking circle $ C_2 $ with radius $ r $ and center the origin. $ P $ is the point $ (0,r) $, $ Q $ is the upper point of intersection of the two circles, and $ R $ is the point of intersection of the line $ PQ $ and the $ x $-axis. What happens to $ R $ as $ C_2 $ shrinks, that is, as $ r \to 0^+ $?

$(4,0)$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

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This is problem number sixty six of the Stuart Calculus eighth edition, Section two point three. The figure shows a fixed circle C one with equation quantity X minus one squared plus y squared equals one and a shrinking circle C two with Radius R and center at the Origin. P is the point zero. Come on. Our Q is the upper point of intersection of the two circles, and our is the point of intersection of the line P Q and the X axis. What happens to our C two shrinks? That is, as our Burt zero from the right. So here we have another o. R. This is Di figure that these problems referencing where we have the fix Circle C one. We have the shrinking circle, too, which is currently centred about the origin. And it'LL remain senator about origin but has a radius of are at this point, and we see that there's language through P and Q. And where do they intersect? Sex excess, that is the point are so to first approach this problem, we should understand what we're Q lies at any point while this circle is shrinking, so we need to find what where Point Q is by equating the equation of the Circle sea, too, with the equation of the Circle Sea one. The equation would foresee. One circle si won has given. However. The equation for Setu is not given, but we can write that because it is centered at the origin. C two has an equation equal to X squared plus y squared equals R squared. So now that we have the equation for both circles, we will seek to sell for the intersection. Cue by equating these two equations, we will first do that by taking this equation and subjecting this equation. And so, for the ex co ordinate, it's a little out of X minus one quantity squared minus X squared. Why square minus what I scored. A zero equals one minus r squared. Here we expand. This term becomes X squared menace to X plus one minus X Word equals to one minus r squared and we see here that X squared X squared the negatives were cancelled plus one on one side and plus one of the others have also cancelled. And if we solve for X, we should get one half r squared. We take this X value, Plug it into the equation of C two in order for us to easily calculate Thie. Why coordinate of Q one f r squared his ex square. This plus y squared equals R squared. This is one fourth art of the fourth plus y squared equals R squared. That means that why squared equals R squared minus one fourth. Aren't you the fourth? If we rearrange the right side of this equation, we're able to affect her and an r squared and we're left with one minus one fourth r squared and then we take the square root. I'm good science and we get our times the square root of the quantity one minus one fourth R squared. So at this point, we should just confirm that Q. The point for Q is one half R squared for X, and then the white coordinate is our time to square root of one minus one fourth R squared. Okay, so we will use what we know about Q and we won't know what we know about Pee Pee is at zero r Q as thes coordinates. And one thing we could do if you could soften the slope of the line that passes through P Q. It will be the difference in the Y values. So are times the square root of one minus one fourth R squared. This one is the wine value of pointy are And then we're dividing by the X value of Q one out of four squared minus the X value of peat, which is zero minus zero. And this is our stop. We will take this slope and use it in a point slope formula where the point we're in uses ur are so why minutes are equals the slope given my em which we just found our times squared of one minus one fourth R squared minus r All the ready by one half are squared, multiplied by x times the quantity x minus the expelling of p which is zero. Okay, and now that we have the equation of this line, we want to understand where this line crosses thie X axis. And that is for why equals zero. So we're going to do is we're going to set y equals zero and figure out a function for X. Ah, that is a function of just are. So we will have the coordinates of the X intercept of this of this lying peak, You and we will then take the limit Is that our approaches the room to figure out where that x point will lie that X intercept to align. So if we re re arranged this, we should can't x equals negative are multiply it by one f r squared, divided rhyme r times or and this numerator part that will be that the denominator on the other side we'LL be able to factor out and our our times this quantity square root of one minus one fourth R squared minus one And at this point, we can make a small simplification canceling out thes ours on our next step will be to rationalize this. I'm also playing the top on the bottom by the A conjugated the bottom. So we're going to open up any page to continue our work. This moment we have this one. We have X equals negative. Ah, one half are squared, divided by our tenets of quantity Square one minus one fourth R squared minus one and more behind by the conjugated thiss radical at the bottom. Ah, what? Actually we do not have that are anymore. So let me just remove that. That was canceled with our animator as we saw in the previous step. Now we want to talk about the country It square root of one minus one fourth R squared plus one. Yes, we all replied to the top and the bottom one minus one for us. R squared plus one. We will get in the numerator Negative one half r squared, you know, to find about this quantity one minutes one fourth R squared plus one from the bottom because we chose the conjure it We should be able to normal or rationalize this denominator. Ah, the radical sign will disappear. We'LL be left with what was inside one when this one fourth r squared Ah, If we forget about the rest, it will cancel except for the last term which is thinking of one times positive one which is minus one this one and this negative one will cancel here received that there's a negative in the numerator and then they're negative in the denominator. Those will cancel, um are squares will cancel and we have one have to right a white one fourth which is just too. So our X function that we have solved for is now in a very simple form. Two times the square root Ah, one minus one fourth R squared plus one. Now, this is the function that we will be taking the limit. The arm perches zero from the right because now we are capable of doing that. So, with direct substitution, we'LL get to what? A pine by the square root of one plus one which is equal to two times two or four. So in the end, well, we have concluded is that as a circle C two shrinks all the way down to R zero all the way until has zero radius. Um, assuming that we are reaching that, why the X intercept, which is where y equals zero. That was the condition that we did all this work here we should get. We should approach an X value of four betweens that this is the point, Cynthy eccentricity on our final answer for the location of our after these circles, he too, has shrunk down to articles era

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