a. Find the radii of the circles x^2+y^2=25 and (x-9)^2+(y-12)^2=100 b. Find the distance betwee

A. Find the radii of the circles x^2+y^2=25 and...

Key Concepts

Standard Equation of a Circle

A circle in the coordinate plane is typically represented by the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius. This form is crucial for identifying both the center’s location and the size of the circle from the equation.

Determining the Radius from the Equation

Once a circle's equation is in its standard form, the radius is obtained by taking the square root of the constant on the right-hand side. This conversion from the squared term to the actual radius is fundamental in understanding the size of the circle.

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the distance between two points in the coordinate plane. It is essential for finding the separation between the centers of two circles, which is given by the formula d = ?[(x? - x?)² + (y? - y?)²].

Tangency Conditions for Circles

For two circles, being externally tangent means they touch at exactly one point without overlapping. This happens when the distance between their centers is equal to the sum of their radii. Understanding these tangency conditions is important for analyzing the relative positions of circles in the plane.

Graphing Circles in the Coordinate Plane

Graphing a circle involves plotting its center point and using the radius to draw a circle that encompasses all points at that distance from the center. This visualization technique helps in understanding the spatial relationships between different circles, including conditions of tangency or intersection.

Transcript

00:03 Okay, in this exercise, we are given two different circles, the one there in black, x squared plus y squared equals 25, and then in red, x minus 9 quantity squared plus x minus 12 quantity squared equals 100.

00:17 The first thing that we're asked to do, we're asked to do multiple things here, the first thing that we're asked to do is to find the radii of the circles or the radius measure for each circle.

00:26 Notice that i've got up there in blue the general form for any circle, and we can tell from that that in the case of the black circle here, remember the center point is usually hk.

00:38 In this case, there is no h, there is no k.

00:41 Those would effectively be zero.

00:43 So the center point here is at zero, zero, and the radius would then be the square root of this number.

00:50 So this is r squared, 25 is r squared, so r would be the square root of 25.

00:56 Or just five.

00:57 So the radius for the first circle is five.

01:00 Similarly, in the second circle, we can kind of do the same thing.

01:04 We can say, okay, the center point here would be 9, 12, and the radius would be the square root of 100, or the radius in this case is 10.

01:14 So the first thing we're asked to do is identify the two radii, and we've done that.

01:19 The second thing we're asked to do is to find the distance between the centers of the circle.

01:23 So we've now noted what the coordinates for the center points are, we can find the distance between those using the distance formula.

01:31 So take the distance is the square root, and i'll say what's the difference between the x's? the difference between the x's is 9 minus 0, square that, and then take the difference between the y's and square that.

01:44 And this is going to give us the square root of 81, 9 squared.

01:49 Maybe i'll write it that way first.

01:51 So we have a nine squared here, nine minus zero, and then 12 squared.

01:59 So essentially that's the square root of 81 plus 144, and that's the square root of 225.

02:08 When we get the square, take the square to 25, we get 15.

02:11 So the distance between the two center points is 15 units.

02:17 Third thing that we're asked to do is explain why the circles must be externally tangent, and then the fourth thing we're asked to do is graph the circles.

02:27 I think if we graph the circles, it's clear a little bit to see why they must be externally tangent.

02:33 So i'm going to do that first.

02:34 I'm going to switch to this other page here.

02:36 And what i've done is i took our two equations.

02:38 So remember, the first circle had an equation of x squared plus y squared equals 25.

02:46 And i'm going to let that be this black circle.

02:48 Over here on my grid, right? i'm going to mark this point here just for illustration purposes.

02:54 I'm going to mark that as the center or sorry, not the center, but the origin.

02:58 And that would actually be the center for this circle.

03:01 So i'm going to center this circle as best i can.

03:04 Let's just take this circle.

03:06 We're going to drag it over here and center it.

03:08 I can tell where it's centered because i'll have these marks here that mark the halfway point.

03:12 So i'm going to center that circle as best i can, right? out there over zero zero...

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