Equations of Circles: Finding Center and Radius

Equations of Circles

Standard Form of a Circle

The equation of a circle in the coordinate plane is commonly written in standard form as:

Standard Form:

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

• (h, k) is the center of the circle.

• r is the radius of the circle.

Identifying Center and Radius

Given the equation:

$(x - 6)^2 + (y + 7)^2 = 49$

• Center: Compare with the standard form. The term $(x - 6)$ means $h = 6$. The term $(y + 7)$ can be rewritten as $(y - (-7))$, so $k = -7$.

• Radius: The right side of the equation is $r^2 = 49$, so $r = \sqrt{49} = 7$.

Summary Table

Feature	Value
Center	(6, -7)
Radius	7

Example

Example: For the equation $(x - 2)^2 + (y + 5)^2 = 16$, the center is (2, -5) and the radius is $\sqrt{16} = 4$.

Applications

• Finding the center and radius is useful for graphing circles and solving geometric problems in algebra.

• Circle equations appear in analytic geometry, physics, and engineering contexts.