The figure shows a fixed circle c1 with equation x 12 y2 1...

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^+ $?

Transcript

0:00 All right.

00:01 So we have these two circles and we want to know as this circle shrinks what happens to a line through two points.

00:08 So so the two points were given or the first one is zero comma are the 2nd 1 is wherever these two circles intersect.

00:19 Mhm.

00:19 Now one of the equations in the red circle, i'm gonna multiply out x minus one squared.

00:25 Sure.

00:26 Plus one plus y squared equals one.

00:29 So the equation that circles x squared minus two, x plus y squared equals zero.

00:34 The equation of the bottom circle is x squared plus y squared equals r squared.

00:39 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.

00:48 And we get that two x equals r squared.

00:52 So x equals r squared over to where the circles intersect.

00:56 Now we're gonna go to the blue equation and plug in r squared over two for x and solve for y.

01:04 So we get y equals the square root of r squared minus r squared over two squares.

01:15 So now we know that point is i'm gonna call this y one and x one x one y 1.

01:22 I just want to have to rewrite all that.

01:24 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.

01:40 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.

01:53 If it were extended meaning we need to set the y equal to zero in the line and solve for x.

02:03 We get negative are um times x one all over why one minus r is equal to x.

02:14 And that x value is actually that point or that we care about.

02:17 Yeah.

02:19 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.

02:29 So let's put everything back in.

02:31 It's gonna be super fun.

02:33 We get negative r times r squared over two over route r squared minus are to the fourth over four minus r...

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