Equation of a Circle

A circle is every point at distance $r$ from a centre $(a, b)$, and writing that with the distance formula gives the standard form:

$(x - a)^2 + (y - b)^2 = r^2$

Expanded, it becomes the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, with centre $(-g, -f)$ and $r = \sqrt{g^2 + f^2 - c}$. This topic is new for 2025-2027, old past papers don’t contain it, and examiners reliably probe new content.

Building the equation, the three standard setups

Given centre and radius: direct substitution. Centre $(3, -2)$, radius $5 \to (x - 3)^2 + (y + 2)^2 = 25$. Watch the double sign-flip: the $-2$ becomes $+2$ inside the bracket, and the right side is $r^2$, not $r$.

Given centre and a point on the circle: $r^2 = \text{distance}^2$ from centre to point, and since the equation wants $r^2$, skip the square root entirely:

Centre $(1, 4)$, through $(4, 8)$: $r^2 = 3^2 + 4^2 = 25 \to$ $(x - 1)^2 + (y - 4)^2 = 25$

Given the endpoints of a diameter: centre $=$ midpoint of the endpoints; $r^2 = \text{distance}^2$ from that centre to either endpoint. Two formulas, then substitute.

Each ingredient (centre, $r^2$) earns its own mark, show the midpoint/distance working rather than announcing the equation.

Moving between forms

Standard to general: expand and collect (rarely asked, easy). General to standard: complete the square twice, the direction the exam loves, covered in detail in centre & radius.

Checking membership and positions

“Show the point $(7, 1)$ lies on the circle”: substitute, show LHS $= r^2$, conclude in words. “Inside or outside?”: compare $\text{distance}^2$ from the centre against $r^2$, less means inside, more means outside; the comparison statement is the answer the command word wants.

Common mistakes

• Centre signs flipped reading $(x + 3)^2$ as centre $+3$
• $r$ used where $r^2$ belongs on the right side
• Square-rooting $r^2$ unnecessarily, then squaring again (surd-handling errors for free)
• Diameter setups using an endpoint as the centre
• “Lies on the circle” verified numerically but never stated

Full topic context: Circle Geometry notes.