You've seen the basic steps for finding the equation of a circle if you are given its graph or key information about its parts. You've also seen how to find the graph (or part of the circle) if you are given the equation. The general equation of a circle makes it very easy for you to solve these types of problems.
So you may be wondering Is that all there is to it? What's the catch? We've already discussed some strategies for figuring out the radius if it's not given, such as applying the distance formula using the center and a point on the circle, or simply dividing the diameter in half. One other potential challenge is if you're given the equation of the circle in a form other than the general equation. In these cases, you'll need to call on your algebra skills.
The video below will demonstrate how to start with a different equation and turn it into the friendlier standard form for the equation of a circle.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
It's nice when an equation of a circle is written in general form. We can easily identify the center point and radius of the circle. However, we may not be given the general form of the equation; rather we might be given an equation that looks like a polynomial with two variables. I'd like to show you a method that can be used to re-write an equation into the general form of a circle using a technique called "Completing the Square."
Our goal is to isolate two variables and create perfect square trinomials that can be factored into a squared binomial. Hopefully, you feel comfortable with your Algebra skills, you will need them for this work.
First we will isolate the constant value by adding it to both sides of the equation. Then, I would like to rearrange the terms of the polynomial. We are allowed to do this because of the commutative property of addition – this means we can add numbers in any order we like without changing the sum. When I rearrange these terms though, I would like to leave some room in the equation to complete each square trinomial. To complete a square trinomial, we must look at the coefficient in front of the linear variable (the variable without an exponent showing). We will divide this coefficient by two and square the result. In this case, we need half of six, and half of eight. We will divide this coefficient by two and square the result. Each of these results are 9 and 16 respectively. These numbers are to be added to both sides of the equation in order to keep the equation balanced.
Square trinomials are especially nice because they can be factored into two identical binomials like this… and we know that the product of two identical values can be written as each factor squared… By using appropriate Algebra techniques, we now have an equation that is written in the general form of a circle. The center of this circle is located at the point -3, -4 and the radius is 7 units.
Question
What are the center and the radius of the circle from the video?
