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In $20-27$ : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle.$$x^{2}+y^{2}-6 x+2 y-6=0$$

(a) $(x-3)^{2}+(y+1)^{2}=16$(b) $(3,-1)$(c) $4$

Algebra

Chapter 4

RELATIONS AND FUNCTIONS

Section 9

Circles

An Introduction to Geometry

Functions

Linear Functions

Polynomials

Missouri State University

Baylor University

Idaho State University

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All right, we are given the equation X squared plus y squared minus six x close to why minus six equals zero. And the first thing I want to do is, um, group the like variables together. So we're gonna put our exes together and put our wise together. So x squared minus six x in a group. And, um, we got a positive life square in a positive to why? So put those in a group in, um, we're gonna be equal to six because we're just gonna add that six over. And there's something called Completing the Square That's gonna help us get to the centre radius form of the equation of a circle. And when we complete, the square will have X squared minus six x. And then we have to come up with 1/3 term that will allow us to write this as a perfect square by Tomio. Same to you with the wise. Okay, so I'm just putting little slot spots for us to figure out what never should go in there. So it's illegal of six. How don't come up with the number to complete the square here. It's from the what will be the middle term here. Negative six if you take half of it, which is negative three and square it negative. Three squared is positive. Nine that it completes the square. That phrase comes from less. Allowing is to write. This now is a perfect square which will do in a moment if we do this with the wise. Now we take half of the middle term to which is one on ones where it is one. Now I'm adding the nine and adding the one to the left side of the equation. We have to state balance. So you have to do this to the right side of the equation as well. So we'll add the nine and the one to the six. All right, so now the first group, um X squared minus six ex post nine could be rewritten as X minus three squared exponentially squared. If you multiply this out is the same thing as X squared minus six X plus nine. A little shortcut forgetting here is it is the number that we're taking 1/2 of in order to square Is that number. We'll do this for the wise as well. Same thing as a Y plus one squared. And if we add the 69 and one together, we get 16. Now we are in Senate Radius form of the equation of a circle That is your answer to part A. The second part of the question asked, Now that we have this form, what is the center? And then what is the radius? From this form? We can see that the center is at three negative warn opposite signs, and we're always equal to R squared for the equation of a circle. So what squared is equal to 16 four? So three negative one is the center for is the radius.

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