a. Find the radii of the circles x^2+y^2=2 and (x-3)^2+(y-3)^2=32 b. Find the distance between

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

Key Concepts

Graphing Circles

Graphing circles in the coordinate plane involves plotting the center based on its coordinates and drawing the circle using its radius. Mastery of graphing is necessary to visually interpret and analyze geometric relationships and tangency between circles.

Internal Tangency

Internal tangency occurs when two circles touch at exactly one point, with one circle lying entirely inside the other. The condition for internal tangency is that the distance between the centers is equal to the difference of the radii. Understanding this concept helps in analyzing circle interactions and placements.

Center and Radius Extraction

Extracting the center and radius from a circle's equation involves identifying the values of h, k, and r from the standard form. This process is essential for understanding the circle's position and size in the coordinate plane.

Equation of a Circle

The standard form equation of a circle is (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. Recognizing and rewriting equations in this form is fundamental when analyzing the properties of a circle.

Distance Formula

The distance formula, given by d = ?[(x? - x?)² + (y? - y?)²], is used to compute the distance between two points, such as the centers of circles. This formula is critical in determining the relative positions of circles and is also used in verifying tangency conditions.

Transcript

00:03 Okay, in this question, we have been asked to identify several things.

00:08 We're given two equations of two circles.

00:11 So we've got a black circle here and a red circle.

00:14 And we've been asked to, first of all, find the radii of the two circles.

00:18 And i've got the blue in blue there.

00:20 I've written the standard form for the equation of a circle.

00:23 So from that, we can take this black circle.

00:26 We can say, okay, well, the center of that circle would be at h comma k.

00:30 But in this case, there is no h or k that those.

00:33 Are both zero.

00:34 So the center of this circle is zero.

00:37 And if r squared is equal to two, then that must mean that r is equal to the square root of two.

00:45 Now i'm going to leave that as a square root of two.

00:48 But i'm also going to go on to my calculator for a minute.

00:50 It'll make more sense in a second why i'm doing this.

00:53 This is about, this is approximately 1 .4 roughly.

00:58 I want to leave the radius there in a square root of two.

01:01 So when i ask me for the radii, i would just report as the square root of 2.

01:05 In the red circle, we would do something similar.

01:07 In this case, we can see that the center point here is going to be 3 ,3, and the radius would be the square root of 32.

01:17 Now, one interesting thing that we might do here is we might say the square root of 32 is the square root of 16 times 2.

01:27 And that would be equal to the square root of 16 times the square root of 2, which would be, four square roots of two.

01:36 Again, why i went through all that's going to become a little bit more clear as we answer another question here in a second.

01:43 Now, again, also i'm going to say that's roughly, it's not perfectly, but that's about 5 .7 thereabouts.

01:51 All right, so let's keep that in mind.

01:52 We've got the two radii, right? we can say the square root of 32, or we can say four square roots of two for the red circle, square root of two for the black circle.

02:00 And then we're supposed to find the distance between the centers of the circle.

02:04 So we can find the distance between these two points pretty quickly.

02:08 We can say, well, the distance will be the square root.

02:11 Find the difference between the two x coordinates, squared.

02:14 That's going to be three minus zero.

02:16 And then the difference between the y coordinates in this case is also three minus zero.

02:20 So we have the square root of three squared plus three squared or the square root of nine plus nine.

02:26 That's the square root of 18.

02:29 Again, we might say, okay, well, square root of 18, that's the square root of 9 times 2, or the square root of 9 times the square root of 2, which is the 3 times the square root of 2.

02:43 Now, that's going to give us the distance between the centers.

02:47 The next two questions.

02:48 The next question asks us a question about why the two circles must be internally tangent.

02:54 And that's maybe not entirely clear right now.

02:57 It will be, if it's not clear right now, it'll be in a second...

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