Centre & Radius

Given a circle equation in expanded form, the exam wants its centre and radius, and the safest extraction is completing the square twice, once in $x$ and once in $y$.

The routine

$x^2 + y^2 - 8x + 6y + 15 = 0$ Group: $(x^2 - 8x) + (y^2 + 6y) = -15$ Complete each square: $(x - 4)^2 - 16 + (y + 3)^2 - 9 = -15$ $(x - 4)^2 + (y + 3)^2 = 10$ Centre $(4, -3)$, radius $\sqrt{10}$

Both completed squares are method marks; the read-off is the accuracy. Leave the radius as a simplified surd on Paper 1, $\sqrt{10}$ is the answer, $3.16$ is a degraded copy.

The general-form shortcut, centre $(-g, -f)$, $r = \sqrt{g^2 + f^2 - c}$ from $x^2 + y^2 + 2gx + 2fy + c = 0$, gives the same results faster if you halve the printed coefficients correctly ($2g = -8 \to g = -4 \to$ centre $x = 4$). Under pressure, completing the square is self-auditing; the shortcut is memory-dependent. Know both, trust the first.

Two traps built into the form

The sign flip: $(y + 3)^2$ means the centre’s $y$-coordinate is $-3$. The $r^2$ trap: the equation’s right side is $r$-squared; “state the radius” wants $\sqrt{\ }$ of it, while “state $r^2$” (feeding an area $\pi r^2$, say) wants it untouched, read which.

A third, sneakier check: a “circle” equation needs $g^2 + f^2 - c > 0$. “Explain why $x^2 + y^2 - 2x + 4y + 9 = 0$ is not a circle” $\to$ completed squares give RHS $= -4 < 0$; no real points, the stated reason is the whole answer.

Centres from geometry, not algebra

When no equation is given, centres come from circle properties:

• From a chord: the centre lies on the perpendicular bisector of every chord; two chords $\to$ two bisectors $\to$ solve simultaneously $\to$ centre
• From a diameter: centre = midpoint of the endpoints
• From a tangent: the centre lies along the perpendicular to the tangent at the contact point (tangent properties)

Common mistakes

• Coefficients halved wrongly using the $(-g, -f)$ shortcut
• Sign flips on the centre coordinates
• Radius reported as $r^2$ (or vice versa)
• Surd radius decimalised
• The existence condition (positive RHS) never checked when the question hints at it

Full topic context: Circle Geometry notes.