Finding a Circle's Centre and Radius

The circle topic (new to the 2025–2027 syllabus) leans on a skill you already have: completing the square, now done twice, once in $x$ and once in $y$. That converts the expanded equation into the standard form $(x - a)^2 + (y - b)^2 = r^2$, which reads off the centre and radius directly.

A circle has equation $x^2 + y^2 - 6x + 4y - 12 = 0$. Find the coordinates of its centre and the length of its radius. [5]

The working

Step 1, group $x$-terms and $y$-terms: $(x^2 - 6x) + (y^2 + 4y) - 12 = 0$

Step 2, complete the square in each bracket. Half of $-6$ is $-3$; half of $4$ is $2$: $(x - 3)^2 - 9 + (y + 2)^2 - 4 - 12 = 0 \quad \text{(M1 for } x, \text{ M1 for } y)$

Step 3, collect the constants ($-9 - 4 - 12 = -25$) and move them across: $(x - 3)^2 + (y + 2)^2 = 25 \quad \text{(A1)}$

Step 4, read off the centre and radius from $(x - a)^2 + (y - b)^2 = r^2$: $\text{centre } (3, -2), \qquad \text{radius } = \sqrt{25} = 5 \quad \text{(A1, A1)}$

Where the marks are won and lost

• The centre’s signs are opposite to the brackets: $(x - 3)$ gives $a = +3$; $(y + 2)$ gives $b = -2$. This sign flip is the most common error.
• The radius is $\sqrt{25} = 5$, not $25$. The right-hand side is $r^2$.
• All three subtracted constants ($-9$, $-4$, and the original $-12$) must be gathered before moving across. Missing one changes the radius.

Common mistakes

• Giving the centre as $(-3, 2)$ (signs not flipped).
• Reporting the radius as $25$ instead of $5$.
• Forgetting to subtract the $9$ and $4$ generated by completing each square.

Full method: Centre & Radius notes. See also Equation of a Circle. Topic home: Circle Geometry pillar.