๐ Standard Form of a Circle
The standard form of a circle's equation is super useful because it directly tells you the center and radius of the circle. It looks like this:
$(x - h)^2 + (y - k)^2 = r^2$
Where:
- ๐ $(h, k)$ is the center of the circle
- ๐ $r$ is the radius of the circle
๐งญ General Form of a Circle
The general form is a bit more disguised but still contains all the same information. It looks like this:
$Ax^2 + Ay^2 + Dx + Ey + F = 0$
Where A, D, E, and F are constants.
๐ Standard vs. General Form: A Comparison
| Feature |
Standard Form |
General Form |
| Equation Structure |
$(x - h)^2 + (y - k)^2 = r^2$ |
$Ax^2 + Ay^2 + Dx + Ey + F = 0$ |
| Center Visibility |
Directly visible as $(h, k)$ |
Not directly visible; requires completing the square |
| Radius Visibility |
Directly visible as $r$ (radius = $\sqrt{r^2}$) |
Not directly visible; requires completing the square |
| Ease of Graphing |
Easier to graph directly |
Requires conversion to standard form first |
โจ Key Takeaways
- ๐ The standard form is great for quickly identifying the center and radius.
- โ๏ธ The general form requires more algebraic manipulation to find the center and radius.
- ๐ก Converting from general to standard form involves completing the square.