Find the center and the radius of the circle with the given equation. [ 10.2] x^2+y^2-6 x+4 y=3

Name: Problem 80, Chapter 11: Precalculus: Graphs and Models, A Right Triangle Approach
Uploaded: 2021-07-09T16:01:39-08:00
Duration: 3 min 43 s
Channel: Chris Wojturski
Description: Find the center and the radius of the circle with the given equation. [ 10.2] x^2+y^2-6 x+4 y=3

step 1 All right, for number 80, a little review here. We want to find the center and the radius of tha

Find the center and the radius of the circle with the given...

Key Concepts

Standard Equation of a Circle

The standard equation of a circle in the plane is typically written as (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents its radius. This form is essential because it directly shows the geometric properties of the circle, making it clear how far points on the circle are from the center.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a perfect square form. This method is crucial when rewriting a circle’s equation from its general quadratic form to its standard form, as it allows you to isolate terms involving x and y and reorganize them into perfect square trinomials that reveal the center and the radius.

Transcript

00:01 All right, for number 80, a little review here.

00:07 We want to find the center and the radius of that circle.

00:11 Now, if you remember this, great, but this is going to require us to do some completing of the square.

00:18 So hopefully you're ready to do a little bit of algebra here.

00:21 Okay.

00:21 We want to put this circle in standard form, which you might recall looks like this.

00:27 X minus h squared plus y minus k square equals r squared.

00:32 And then, of course, the center is going to be the point hk, and, of course, the radius is r.

00:40 So whatever you get on the right, you just square root it, and that's your radius.

00:44 So let's recall how to do this a little bit.

00:47 I'm going to rewrite the terms.

00:49 It might look a little bit strange, but for those of you that remember how to complete the square, maybe this will make some sense to you.

00:56 So i'm going to do x squared minus 6x, and then i'm going to leave a space.

01:02 Then i'm going to put plus y squared, then the y term, 4y, leave a space, and then i set it equal to 3.

01:11 And in those spaces here, this is where we're going to complete the square.

01:14 Remember what that means is to find a quantity that will allow us to factor it into two of the same thing.

01:23 And it's really easy to do.

01:24 If you don't remember the process, here you go.

01:26 You're going to take the, oops, i'm going to do that, sorry.

01:30 You're going to take the six.

01:33 Don't worry that it's negative because after you take half, you square it...

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