The figure shows a fixed circle C1 with equation (x

Key Concepts

Line Equations and Intersections

Once two points are known, the concept of the line that passes through them becomes important. This involves calculating the slope between the points and determining the line's equation in slope-intercept or point-slope form. Additionally, finding the intersection of this line with another geometric object like an axis is a key application of coordinate geometry.

Limit Analysis in Geometry

Limit analysis in geometry is used to study the behavior of geometric figures as some parameters approach a specific value, such as zero or infinity. This technique allows the determination of asymptotic behavior or the ultimate position of geometric constructs when a certain element of the configuration changes, which is essential in understanding dynamic geometric systems.

Circle Geometry

Circle geometry involves understanding the algebraic representation of circles using equations of the form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This concept is fundamental in analyzing the properties of circles, such as their location, size, and how they interact with other geometric objects.

Intersection of Circles

The problem of finding the intersection points of two circles is a classical topic in analytic geometry. It requires solving a system of equations, which leads to determining the coordinates that satisfy both circle equations simultaneously. This process is crucial for understanding how two circles relate to each other and for defining additional geometric constructs based on their intersections.

Transcript

00:01 This is problem number sixty six of the stuart calculus eighth edition, section two point three.

00:07 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.

00:21 P is the point zero.

00:23 Come on.

00:23 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.

00:32 What happens to our c two shrinks? that is, as our burt zero from the right.

00:37 So here we have another o.

00:39 R.

00:40 This is di figure that these problems referencing where we have the fix circle c one.

00:48 We have the shrinking circle, too, which is currently centred about the origin.

00:55 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.

01:03 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.

01:27 The equation would foresee.

01:28 One circle si won has given.

01:30 However.

01:31 The equation for setu is not given, but we can write that because it is centered at the origin.

01:37 C two has an equation equal to x squared plus y squared equals r squared.

01:49 So now that we have the equation for both circles, we will seek to sell for the intersection.

01:57 Cue by equating these two equations, we will first do that by taking this equation and subjecting this equation.

02:05 And so, for the ex co ordinate, it's a little out of x minus one quantity squared minus x squared.

02:16 Why square minus what i scored.

02:17 A zero equals one minus r squared.

02:24 Here we expand.

02:25 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.

02:45 And if we solve for x, we should get one half r squared.

02:50 We take this x value, plug it into the equation of c two in order for us to easily calculate thie.

02:57 Why coordinate of q one f r squared his ex square.

03:03 This plus y squared equals r squared.

03:09 This is one fourth art of the fourth plus y squared equals r squared.

03:19 That means that why squared equals r squared minus one fourth.

03:25 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.

03:45 I'm good science and we get our times the square root of the quantity one minus one fourth r squared.

03:53 So at this point, we should just confirm that q.

03:59 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.

04:20 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...

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