Part 4/5:
The squared lengths give us the equation ( (y - b)^2 + (x - a)^2 = r^2 ), affirming our original circle equation.
Example of a Circle's Equation
As a practical example, consider a circle with a radius of ( 4 ) centered at the origin (0,0). Utilizing the general form, we can derive:
[
x^2 + y^2 = 4^2 \implies x^2 + y^2 = 16
]
This simple exercise illustrates how circles can be represented mathematically, while simultaneously reinforcing the conceptual understanding of their geometric properties.