Circle Equations in the Coordinate Plane

Standard Form

$(x - h)^2 + (y - k)^2 = r^2$

• Centre: $(h, k)$
• Radius: $r$

General Form

$x^2 + y^2 + Dx + Ey + F = 0$

To find centre and radius, convert to standard form by completing the square.

Completing the Square for Circles

Example: $x^2 + y^2 + 6x - 4y - 12 = 0$

Group: $(x^2 + 6x) + (y^2 - 4y) = 12$

Complete the square:

• $x$: $(x^2 + 6x + 9) = (x+3)^2$. Added 9.
• $y$: $(y^2 - 4y + 4) = (y-2)^2$. Added 4.

$(x+3)^2 + (y-2)^2 = 12 + 9 + 4 = 25$

Centre: $(-3, 2)$. Radius: $5$.

Points on a Circle

A point $(a, b)$ is on the circle if it satisfies the equation.

Tip 1: Standard Form Gives Centre and Radius Directly

$(x-3)^2 + (y+1)^2 = 16$ → Centre $(3, -1)$, radius $4$.

Tip 2: Complete the Square Systematically

Group $x$ terms and $y$ terms. Add the square-completion values to both sides.

Tip 3: Remember $r^2$, Not $r$

The right side of the standard form is $r^2$. To find $r$, take the square root.