Write the equation of a circle in standard and general form and find the centre and radius by completing the square

What this dot point is asking

SEAB wants you to write the equation of a circle from its centre and radius, to recognise the general expanded form, and to recover the centre and radius from that general form by completing the square. The circle is the one curve beyond the straight line in O-Level coordinate geometry, and these manipulations are the core of every circle question.

The answer

The standard form

A circle of centre $(a, b)$ and radius $r$ has the equation:

This is just the distance formula: every point on the circle is a distance $r$ from the centre. Read the centre and radius straight off.

The general form

Expanding the standard form gives the general equation:

where the centre is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$ (provided this is positive). The hallmarks of a circle equation are equal coefficients of $x^2$ and $y^2$ and no $xy$ term.

Recovering centre and radius

To go from the general form to the centre and radius, complete the square in $x$ and in $y$ separately, then move the constant across. The completed-square brackets give the centre; the right-hand side gives $r^2$.

Building the equation from data

If you know the centre and a point on the circle, the radius is the distance between them. If you know the endpoints of a diameter, the centre is their midpoint and the radius is half the diameter's length.

Testing whether a point lies on the circle

Substitute the point into the equation: if it satisfies $(x - a)^2 + (y - b)^2 = r^2$, the point is on the circle. If the left side is less than $r^2$ the point is inside, and if greater it is outside, which is a quick way to classify a location relative to a circular boundary.

The diameter and a right angle

A useful circle fact is that an angle in a semicircle is a right angle: if $AB$ is a diameter and $P$ is any other point on the circle, then angle $APB$ is $90^\circ$. This connects the circle equation to perpendicular-gradient checks from the straight-line topic.

Examples in context

Example 1. Range of a transmitter. A radio mast covering all points within a fixed distance defines a circular boundary; its equation, centred on the mast, tells you instantly whether a given location lies inside the coverage by comparing distances.

Example 2. Designing a circular track. Given three points a track must pass through, completing the square on the general equation found from those conditions recovers the centre and radius, the practical design problem behind the algebra.

Try this

Q1. State the centre and radius of $(x - 2)^2 + (y + 5)^2 = 9$. [2 marks]

• Cue. Centre $(2, -5)$, radius $3$.

Q2. Write the equation of the circle with centre $(0, 0)$ and radius $7$. [1 mark]

• Cue. $x^2 + y^2 = 49$.

Q3. Find the centre and radius of $x^2 + y^2 + 2x - 8y + 1 = 0$. [3 marks]

• Cue. $(x + 1)^2 + (y - 4)^2 = 16$, so centre $(-1, 4)$, radius $4$.