Understanding the General Form of a Circle with Center at the Origin

(Keyword: circle equation, general form, center at origin)

The equation of a circle is a fundamental concept in geometry, which can be expressed in different forms. This article will focus on the general form of a circle with the center at the origin and a given radius. We will derive the equation step-by-step and explore how the equation reflects the properties of the circle.

Standard Form of a Circle

The standard form of the equation of a circle is given by:

(x - h)2 (y - k)2 r2

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

Deriving the General Form from the Standard Form

Given a circle with center at the origin (0, 0) and a radius of 16 units:

Start with the standard form of the circle's equation: Substitute the center (0, 0) and the radius 16: Simplify the equation.

Step 1: Substitute the given information into the standard form:

(x - 0)2 (y - 0)2 162

Step 2: Simplify the equation:

x2 y2 256

General Form of a Circle

The general form of a circle with center at the origin can be derived from the standard form. Here’s how it works:

The general form of the circle's equation is:

x2 y2 - 2gx - 2fy - g2 - f2 - r2 0

Where (g, f) are the coordinates of the center and r is the radius.

Applying the General Form to the Given Circle

Given:

Center: (0, 0) Radius: 16 units

The coordinates of the center are 0 and 0, and the radius is 16. Substituting these values into the general form:

x2 y2 - 2g(x) - 2f(y) - g2 - f2 - 162 0

Since the center is (0, 0), g 0 and f 0:

x2 y2 - 2(0)x - 2(0)y - 02 - 02 - 256 0

This simplifies to:

x2 y2 - 256 0

Double-checking the Equation

To ensure the equation is correct, you can substitute a point that lies on the circle into the equation. For example, the point (16, 0) should satisfy the equation:

(16)2 (0)2 - 256 0

256 0 - 256 0

This confirms that the equation is correct.

Conclusion

In summary, the general form of a circle with the center at the origin and a radius of 16 units is: