Equation of a Circle
🎯 What is a Circle?
A circle is the set of all points in a plane that are at a fixed distance (called the radius) from a fixed point (called the center).
This simple geometric definition leads us to one of the most elegant equations in coordinate geometry.
📐 Deriving the Standard Form
Let's derive the equation step by step using the distance formula.
Setup: Let the center of the circle be at point $ C(h, k) $ and let the radius be $ r $. If $ P(x, y) $ is any point on the circle, then:
Distance from $ P $ to $ C $ = $ r $
Using the distance formula:
$$ \sqrt{(x - h)^2 + (y - k)^2} = r $$
Squaring both sides to eliminate the square root:
Standard Form: $$ (x - h)^2 + (y - k)^2 = r^2 $$
Where:
- $ (h, k) $ = coordinates of the center
- $ r $ = radius of the circle (always positive)
- $ (x, y) $ = any point on the circle