The equation of a circle is a mathematical formula used to define all the points that lie on the circumference of a circle in a two-dimensional coordinate system. While the fundamental equations are universal, they are applied and studied in academic institutions worldwide, including those in Cambridge.
The two most common forms for the equation of a circle are the Standard Form and the General Form.
The Standard Form of the Equation of a Circle
The most intuitive way to write the equation of a circle is the Standard Form. It directly shows the center and radius of the circle.
The equation is:
(x - h)² + (y - k)² = r²
Where:
(x, y)represents any point on the circle.(h, k)represents the coordinates of the center of the circle.rrepresents the radius of the circle.
This form is derived directly from the distance formula, as every point on the circle is equidistant (by the radius r) from the center (h, k).
Example:
An equation like (x - 2)² + (y - 3)² = 25 describes a circle centered at (2, 3) with a radius of √25 = 5.
The General Form of the Equation of a Circle
As provided in the reference, the General Form of the equation of a circle is:
x² + y² + 2gx + 2fy + c = 0
This form is obtained by expanding the Standard Form and rearranging the terms. While it doesn't immediately show the center and radius, these can be determined from the coefficients g, f, and c.
According to the reference:
- The center of the circle is (-g, -f).
- The radius of the circle is √(g² + f² - c).
It is important to note that for the equation to represent a real circle, the value inside the square root for the radius (g² + f² - c) must be greater than zero. If it's zero, the equation represents a single point (a circle with radius 0). If it's negative, it represents no real locus.
The reference also states that you can "complete the square to change it into the standard form". This algebraic technique is used to convert the General Form back into the Standard Form, making it easy to identify the center and radius.
Example Conversion (from General to Standard):
Consider the general form: x² + y² - 4x - 6y - 12 = 0
Comparing this to x² + y² + 2gx + 2fy + c = 0:
2g = -4=>g = -22f = -6=>f = -3c = -12
Using the formulas from the reference:
- Center =
(-g, -f)=(-(-2), -(-3))=(2, 3) - Radius =
√(g² + f² - c)=√((-2)² + (-3)² - (-12))=√(4 + 9 + 12)=√25=5
This matches the previous example in the Standard Form.
Comparing the Forms
Here's a quick look at the key differences between the two forms:
| Feature | Standard Form | General Form |
|---|---|---|
| Equation | (x - h)² + (y - k)² = r² |
x² + y² + 2gx + 2fy + c = 0 |
| Center | (h, k) |
(-g, -f) |
| Radius | r |
√(g² + f² - c) |
| Usefulness | Directly shows center/radius; useful for graphing | Useful for algebraic manipulation; sometimes given initially |
In academic settings like Cambridge, students learn to work with both forms, converting between them as needed to solve problems related to circles, such as finding intersections with lines, determining tangents, or analyzing geometric properties.
