The General Equation of a Circle
Simple Explanation
A circle's equation is not always given in standard form — it may be expanded out as x²+y²+Dx+Ey+F=0. Completing the square on the x and y terms converts it back into standard form, revealing the center and radius directly.
Why Do We Need It?
Equations that come from calculations rarely arrive pre-packaged in standard form — this technique recovers the geometrically meaningful center and radius from whatever form the equation happens to take.
Formula
The General Equation of a Circle
x² + y² + Dx + Ey + F = 0
A circle's equation, expanded out rather than in standard center-radius form — completing the square on the x and y terms recovers the standard form.
- D, E, F
- — the expanded equation's coefficients
When to use it: Whenever a circle's equation is given in expanded form and its center/radius need to be found.
Worked Example
Find a circle's center and radius by completing the square
Find the center and radius of the circle x²+y²−6x+4y−12=0.
Why Does This Work?
Completing the square is a reversible algebraic rearrangement — adding and subtracting the same constant does not change the equation's solutions, so the resulting standard-form equation describes exactly the same circle as the original expanded one.
Real-Life Example
Recovering a Wi-Fi router's coverage circle from a signal-strength equation
Engineers derive an expanded equation for a router's coverage boundary from raw signal-strength measurements.
Completing the square converts this into standard form, revealing the router's actual position (center) and coverage radius directly.
Practice
Find the radius of the circle x²+y²+2x−4y−4=0, after completing the square.
MediumCommon mistake
Adding the completing-the-square constants only to the left side of the equation — the exact same constants must also be added to the RIGHT side, or the equation's meaning changes.
Quick Review
- Group terms by variable, then complete the square for each one separately.
- Add the same constants to both sides of the equation.
- The result reveals the center (h,k) and radius r directly.