Circle Standard and General Equations: Concepts and Examples

A circle is the set of points at a fixed distance (the radius) from a fixed point (the centre). In standard form (x − h)² + (y − k)² = r² the centre (h, k) and radius r are readable at a glance. In general form x² + y² + 2gx + 2fy + c = 0 the same circle is written with the coefficients of x² and y² equal to 1 and no xy term; completing the square recovers the centre and radius.

This matters because exam questions hand you the general form and ask for the centre, radius, or whether a point lies inside, on, or outside the circle. Converting between the two forms quickly — and knowing the radius formula sqrt(g² + f² − c) — is the gateway to every harder circle topic: tangents, chords, and coaxial families.

Key Concepts

• Standard form: (x − h)² + (y − k)² = r² — centre (h, k), radius r.
• General form: x² + y² + 2gx + 2fy + c = 0 — centre (−g, −f), radius sqrt(g² + f² − c). Note the sign flip on the centre coordinates.
• Reality condition: the equation represents a real circle only if g² + f² − c > 0; zero gives a point circle, negative gives nothing real.
• Point position: for P(x1, y1), the value S1 = x1² + y1² + 2gx1 + 2fy1 + c is negative inside, zero on, and positive outside the circle.

Formulas

• Standard: (x − h)² + (y − k)² = r²; General: x² + y² + 2gx + 2fy + c = 0.
• Centre and radius from general form: (−g, −f), r = sqrt(g² + f² − c).
• Units: g, f, c carry length, length, and length² units respectively so that every term is an area.
• Assumptions: coefficients of x² and y² are equal and non-zero, and there is no xy term — otherwise the curve is not a circle.

Worked Example

Find the centre and radius of x² + y² + 6x − 8y + 9 = 0, and check where (1, 4) lies.

1. Match with x² + y² + 2gx + 2fy + c = 0: 2g = 6 so g = 3; 2f = −8 so f = −4; c = 9.
2. Centre = (−g, −f) = (−3, 4).
3. Radius = sqrt(g² + f² − c) = sqrt(9 + 16 − 9) = sqrt(16) = 4.
4. Verify by completing the square: (x² + 6x + 9) + (y² − 8y + 16) = −9 + 9 + 16, i.e. (x + 3)² + (y − 4)² = 16. Same centre and radius.
5. Position of (1, 4): S1 = 1 + 16 + 6 − 32 + 9 = 0. The point lies exactly on the circle (it is 4 units from the centre).

Common Misconceptions

• "The centre is (g, f)." It is (−g, −f) — the signs flip, a constant source of lost marks.
• "Any equation with x² and y² is a circle." The x² and y² coefficients must be equal and the xy term must be absent.
• "Radius is g² + f² − c." That is the radius squared; the radius is its square root.

Real-World Applications

• Engineering drawings: bolt-hole patterns and wheels are specified in centre–radius form for CNC machines.
• Computer vision: the Hough circle transform detects circular objects by voting in (h, k, r) parameter space.
• Navigation: range circles from known stations are intersected to fix a position, exactly as in GPS trilateration.
• Game physics: collision checks between circular hitboxes use the standard form to test overlap in constant time.

In short: standard form shows the geometry directly, while general form hides it behind 2g, 2f, c — centre (−g, −f), radius sqrt(g² + f² − c). Flip the signs, take the square root, and check g² + f² − c > 0 before claiming a real circle.