Circles are essentially a set of points equally distant from a certain point.
You can try graphing a circle using the formula x² + y² = 1. This essentially gives you a circle with a radius of one unit.
What's interesting is that if you change the constant, you can see that the circle's radius scales with it. If you multiply this constant by a certain number, the radius will scale up by the root of said number. So the basic formula of a circle with a center on (0,0) is x² + y² = r²., where r represents the circle's radius.
Here's where it gets interesting.
By using the distance formula √ [(x₂ - x₁)² + (y₂ - y₁)²], we get (x-a)² + (y-b)² = r², where a is the horizontal shift, and b is the vertical shift. Remember that the circle's center also gets translated, so we can determine that the formula for a circle's center point is (a,b). Make sure not to mess up the negative signs!
We can turn this formula into a more general form by expanding it.
(x-a)² + (y-b)² = r²
x² - 2ax + a² + y² - 2by + b² = r² x² + y² - 2ax - 2by + a² + b² - r² = 0 We will create three new variables, A, B, and C.
Let A be equal to -2a, B be equal to -2b, and C be equal to a² + b² - r².
Now we are left with the general form x² + y² + Ax + By + C = 0. Since a circle's center is (a,b), it is also equal to (-½A, -½B). C = a² + b² - r², hence r = √(a² + b² - C), therefore r = √ [(-½A)² + (-½B)² - C].
Uhm, thank you for the math lesson. I love them but I don't always understand them. Sometimes they are a bit too difficult for me. But I am grateful for your lesson. I'll make sure to use this to impress my bandmates with my knowledge.
By the way, why is the radius r? Is it just a symbol or is there a meaning to it?