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Problem 66 Hard Difficulty

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^+ $?



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Video Transcript

All right. So we have these two circles and we want to know as this circle shrinks What happens to a line through two points. So so the two points were given or the first one is zero comma are The 2nd 1 is wherever these two circles intersect. Mhm. Now one of the equations in the red circle, I'm gonna multiply out X minus one squared. Sure. Plus one plus Y squared equals one. So the equation that circles X squared minus two, X plus y squared equals zero. The equation of the bottom circle is x squared plus Y squared equals r squared. So what I'm gonna do to get a piece of information about where those two things intersect, I'm gonna take r squared, I'm gonna replace X squared plus y squared. And we get that two x equals r squared. So X equals r squared over to where the circles intersect. Now we're gonna go to the blue equation and plug in R squared over two for x and solve for y. So we get y equals the square root of r squared minus r squared over two squares. So now we know that point is I'm gonna call this Y one and X one X one Y 1. I just want to have to rewrite all that. Okay, so the equation of our line, the slope is gonna be Why 1- are over X 1 0 times and then X zero plus are using the first point. So the equation online is y equals y one minus are over X one times X plus are Now our equation asks us if there's some point are that's the X intercept of this line. If it were extended meaning we need to set the y equal to zero in the line and solve for X. We get negative are um times X one all over why one minus R is equal to X. And that X value is actually that point or that we care about. Yeah. Okay so now we know something about our we're gonna replace because we need to know what is the limit as our approaches zero of that point are. So let's put everything back in. It's gonna be super fun. We get negative R times R squared over two over route R squared minus are to the fourth over four minus R. I mean you know what happens as our approaches 0? So this doesn't look like a ton of fun. I'm gonna do one thing real quick. Let's say we multiply by the cons you get on top and bottom because usually when you have a lackey looking um radicals and you have a limit that is indeterminant as you approach zero. You can always try to multiply by the conjugate. Yeah, so that would be what happens when we multiply by the conjugate. I did it with the simpler symbols. Just to save my hand from some writing We have negative are times X one which is R squared over two times Y one which is the radical. Okay, mine is R squared times X one which would be Are to the 4th over 4th over then. Why one squared is R squared minus? Are to the fourth over four minus R squared. Okay so continuing trying to hammer through this limit I get it negative. R cubed over to times that radical of R squared minus. Or to the fourth over four. Mine is part of the 4th over four. All over negative part of the 4th over four. So now let's simplify that radical a little bit. Leave the limit as our approaches zero of now this is for R squared minus art of the fourth. All over four so We can pull our over two out. We already have our cubed over three. I'm sorry already cubed over to That gives us thinking part of the 4th over four Times The Square Root of four -R. It is already the 4th over four over Negative or to the 4th over four. Do some algebra Get the limit as our approaches zero All those are the force over four. Just go away so we get the square root four minus R plus one. Sorry I have to go back and make a correction. This is our to the fourth over two because it's X one times R squared. Yes. So when we simplify this portion on the right that turns into it too and then as our tends to zero that equals the square root of four plus two, which is four.