Circle Equation: Finding the Equation of a Circle Given the Center and Radius

The equation of a circle is a fundamental concept in algebra and geometry, often appearing in mathematical problems and applications. When given the center and radius of a circle, we can determine its equation using the standard form. This article will guide you through the process, providing clear explanations and examples.

General Form of a Circle's Equation

The general equation of a circle is given by:

x2 - 2hx h2 y2 - 2ky k2 r2 or (x-h)2 (y-k)2 r2

Here, ((h, k)) is the center of the circle, and (r) is the radius.

Finding the Equation of a Circle with Center ((2, -3)) and Radius 4

Given a circle with center ((2, -3)) and radius (4), we can substitute these values into the general form to find its equation.

Let's break down the steps:

Identify the center and radius: ((h, k) (2, -3)), (r 4). Substitute the values into the equation:

x2 - 2hx h2 y2 - 2ky k2 r2

Substituting (h 2), (k -3), and (r 4), we get:

x2 - 2(2)x (2)2 y2 - 2(-3)y (-3)2 42

Simplifying this:

x2 - 4x 4 y2 6y 9 16

Rearranging terms, we obtain:

x2 y2 - 4x 6y 13 16

Subtracting 13 from both sides:

x2 y2 - 4x 6y 3

This is the standard form of the circle equation.

Alternative Form: General Form

The equation can also be written in the general form:

x2 y2 - 4x 6y - 3 0

This form is useful for other algebraic manipulations and solving related problems.

Conclusion

The process of finding the equation of a circle given its center and radius is straightforward when you follow the steps using the standard form. This article has demonstrated how to derive the equation for a circle with a center at ((2, -3)) and radius (4). Understanding these concepts is crucial for solving various algebraic and geometric problems.

Key Takeaways

The general form of a circle's equation is (x - h)2 (y - k)2 r2. To find the equation, substitute the center ((h, k)) and the radius (r) into the formula. The equation can also be expressed in the standard or general form.

For further practice and more in-depth understanding, explore similar problems and utilize algebraic tools effectively.