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Equations of Circles in Mathematics

Mastering the Standard and General Forms

In Coordinate Geometry, a circle is defined as the set of all points that are a specific distance (radius) from a fixed point (center). Unlike linear equations that produce straight lines, circle equations are quadratic—they involve x² and y².

Understanding the equation of a circle is crucial for solving problems in physics (like satellite orbits), engineering (gears and wheels), and computer graphics.

1. The Definition

A circle is formed by two key components:

  • Center (h, k): The fixed middle point.
  • Radius (r): The distance from the center to any point on the edge.

2. The Standard Equation of a Circle

This is the most useful form because it allows you to instantly see the center and radius.

[Image of circle equation graph]
(x - h)² + (y - k)² = r²
  • (h, k) are the coordinates of the center.
  • r is the radius.

Important Note: Notice the minus signs in the formula. If you see (x - 3), the x-coordinate of the center is +3. If you see (x + 3), the x-coordinate is -3.

Example 1:

Find the center and radius of the circle: (x - 2)² + (y + 5)² = 16

  • Comparing to (x - h), h = 2.
  • Comparing to (y - k), k = -5 (because y - (-5) = y + 5).
  • r² = 16, so radius r = √16 = 4.

Center: (2, -5), Radius: 4

3. Circle Centered at the Origin

If the center of the circle is at the Origin (0,0), then h=0 and k=0. The formula simplifies beautifully.

[Image of circle centered at origin]
x² + y² = r²

This is actually a direct application of the Pythagorean Theorem.

4. The General Form Equation

Sometimes, the equation is expanded and set to zero. This is called the General Form.

x² + y² + Dx + Ey + F = 0

While this looks tidy, it hides the center and radius. To find them, mathematicians use a technique called Completing the Square to convert it back to Standard Form.

5. Writing the Equation from a Graph

If you are given a circle on a graph, follow these steps:

  1. Find the coordinate of the center point (h, k).
  2. Count the grid units from the center to the edge to find the radius (r).
  3. Plug them into the standard formula.

Example: Center at (3, 4) and Radius is 6.
Equation: (x - 3)² + (y - 4)² = 36.

Conclusion

The Equation of a Circle is one of the most elegant formulas in geometry. By mastering the transition between the graph and the equation, you gain control over circular motion and design in the coordinate plane.