Circle Notes: Definition, Center/Radius, and Standard Equation

A circle is the set of all points in a plane that are a fixed distance from a fixed point called the center.
Radius, denoted by $r$ , is the distance from the center to any point on the circle.
Center, denoted by coordinates $(h,k)$ , is the fixed point from which all radii extend.
Circle vs Disc:
- Circle: only the edge (the circumference) consisting of points at distance $r$ from the center.
- Disc: the interior region enclosed by the circle.
In the standard circle equation, $x$ and $y$ are the variables (coordinates of any point on the circle), while $h$ , $k$ , and $r$ are constants (parameters of a specific circle).
Summary of two key pieces of information needed to define a circle: the center coordinates $(h,k)$ and the radius $r$ .
Real-world note: problems may give you either the radius or the diameter (see below) and you must read carefully.

In any given circle, $h$ , $k$ , and $r$ are constant values.
The variables in the equation are $x$ and $y$ , representing coordinates of points on the circle.
Two essential pieces of information to form the equation: the length of the radius $r$ and the center coordinates $(h,k)$ .
The task sometimes is to deduce $h,k,r$ from other given data (e.g., endpoints of a diameter or a point on the circle).

The diameter is the straight line segment that passes through the center and connects two points on the circle.
Diameter length $D = 2r$ , so the radius is $r = frac{D}{2}$ .
When a problem provides the diameter instead of the radius, convert to the radius before forming the equation.
If only a segment through the circle is drawn, remember the diameter is the full line across the circle; the radius is half of that.

The standard (center-radius) form of the circle’s equation is:

(x - h)^2 + (y - k)^2 = r^2
This form states: the locus of all points $(x,y)$ whose squared distances to the center $(h,k)$ sum to the radius squared $r^2$ .
Note: some texts present the equivalent form as:

r^2 = (x - h)^2 + (y - k)^2
Substituting the center coordinates and radius into the formula yields the specific circle.

If you know the center coordinates and radius, plug directly into the standard form.
If you know two points on the circle and the center, you can compute radius using the distance formula:

r = ext{distance}((h,k), (x2,y2)) = \
\sqrt{(x2 - h)^2 + (y2 - k)^2}
If two opposite points on the circle are given (i.e., endpoints of a diameter), the center is the midpoint of those points:

h = \frac{x1 + x2}{2}, \quad k = \frac{y1 + y2}{2}
These tools (distance and midpoint formulas) help deduce $h,k,r$ when data is incomplete.

Given: radius $r = 5$ and center $(h,k) = (3,-7)$ .
Start from the standard form

(x - h)^2 + (y - k)^2 = r^2
Substitute the given values: $h = 3$ , $k = -7$ , $r = 5$ :

(x - 3)^2 + (y - (-7))^2 = 5^2
Simplify the subtraction with a negative center: $y - (-7) = y + 7$
Simplify the radius squared: $5^2 = 25$
Resulting equation (in expanded-friendly form):

(x - 3)^2 + (y + 7)^2 = 25
Alternative arrangement shown in the transcript: $25 = (x - 3)^2 + (y + 7)^2$ (equivalent to the previous form)
Important notes from the example:
- The center coordinates determine the shift of the circle in the plane.
- The radius controls how far points on the circle are from the center.
- When the center’s y-coordinate is negative, the term is moved to the addend: $y - (-7) = y + 7$ .
- In this simple example, the radius was readily given; in homework, you may face more complex scenarios requiring you to compute one or more of $h,k,r$ from other data.
The instructor highlighted that homework problems may introduce curveballs beyond this simple setup.

Always distinguish between circle (the edge) and disc (the interior).
Read problems carefully to determine whether you are given the radius or the diameter; convert diameters to radii when needed:
$r = \frac{D}{2}$
Remember the signs when substituting the center coordinates: subtracting a negative center coordinate increases the value (e.g., $y - (-7) = y + 7$ ).
The equation describes all points (x,y) on the circle; not just a single point.
When solving problems, you can present the equation in either of these equivalent forms:

(x - h)^2 + (y - k)^2 = r^2

or

r^2 = (x - h)^2 + (y - k)^2
Practice makes reading these problems faster: identify the center and the radius first, then plug into the standard form.