Circles are one of the most predictable topics in the O-Level Additional Mathematics (4049) syllabus. Almost every year there is a question worth 6 to 10 marks on coordinate geometry of circles, and the structure repeats: find the centre and radius, then do something with a tangent, a chord, or a line that intersects the circle.
That predictability is good news. Once you drill the four or five standard moves below, circle questions become some of the fastest marks in Paper 1 or Paper 2.
Know Your Two Forms Cold
Everything starts with recognising which form you have been given.
Standard (centre-radius) form:
$$(x-a)^2 + (y-b)^2 = r^2$$
Centre is $(a, b)$, radius is $r$. Watch the signs: $(x+3)^2 + (y-1)^2 = 16$ has centre $(-3, 1)$, not $(3, 1)$.
General form:
$$x^2 + y^2 + 2gx + 2fy + c = 0$$
You convert to standard form by completing the square. Do not memorise the shortcut formula blindly, because examiners often want to see the completed square line, and it also protects you when the coefficients of $x^2$ and $y^2$ are not 1.
Worked Example 1: Centre and Radius
Find the centre and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$.
- Group terms: $(x^2 - 6x) + (y^2 + 4y) = 12$
- Complete the square: $(x-3)^2 - 9 + (y+2)^2 - 4 = 12$
- Tidy up: $(x-3)^2 + (y+2)^2 = 25$
Centre $(3, -2)$, radius 5. Two marks, thirty seconds.
If the equation begins with $2x^2 + 2y^2$, divide the whole equation by 2 first. Students lose easy marks every year by skipping this step.
The Tangent Rule That Unlocks Most Questions
The single most examined fact is this: a tangent is perpendicular to the radius at the point of contact.
So if you are asked for the tangent at a point $P$ on the circle:
- Find the gradient of the radius from the centre $C$ to $P$.
- Take the negative reciprocal for the tangent gradient.
- Use $y - y_1 = m(x - x_1)$ with the point $P$.
Worked Example 2: Tangent at a Point
Using the circle above, show that $P(7, 1)$ lies on the circle and find the equation of the tangent at $P$.
Check: $(7-3)^2 + (1+2)^2 = 16 + 9 = 25$, so $P$ lies on the circle.
Gradient of $CP = \dfrac{1-(-2)}{7-3} = \dfrac{3}{4}$, so the tangent gradient is $-\dfrac{4}{3}$.
Tangent: $y - 1 = -\dfrac{4}{3}(x - 7)$, giving $4x + 3y = 31$.
Lines Meeting Circles: Use the Discriminant
When a question says a line is tangent to a circle, or asks for the range of values for which a line cuts a circle at two points, substitute and use the discriminant.
Worked Example 3: Tangency Condition
Find the values of $k$ for which $y = 2x + k$ is a tangent to $x^2 + y^2 = 20$.
Substituting: $x^2 + (2x+k)^2 = 20$, so $5x^2 + 4kx + k^2 - 20 = 0$.
For tangency, $b^2 - 4ac = 0$:
$$16k^2 - 20(k^2 - 20) = 0 \Rightarrow -4k^2 + 400 = 0 \Rightarrow k = \pm 10$$
A useful alternative: the perpendicular distance from the centre to the line equals the radius. Both methods earn full marks, so pick whichever gives cleaner numbers.
Also remember the interpretation:
- $b^2 - 4ac > 0$: line cuts the circle at two points
- $b^2 - 4ac = 0$: line is a tangent
- $b^2 - 4ac < 0$: line misses the circle
Three Circle Theorems Worth Memorising
These let you skip long algebra, and quoting them earns method marks.
- The perpendicular from the centre to a chord bisects the chord. Great for finding a centre or a radius using Pythagoras.
- The angle in a semicircle is a right angle. If $AB$ is a diameter and $P$ is on the circle, then $AP \perp BP$.
- Endpoints of a diameter: the centre is the midpoint of the two endpoints, and the radius is half the distance between them.
For a circle through three given points, find the perpendicular bisectors of two chords and solve them simultaneously. The intersection is the centre.
Common Mistakes Examiners See Every Year
- Writing the radius as $r^2$, for example claiming radius 25 instead of 5.
- Sign errors reading the centre from standard form.
- Forgetting the second answer when a quadratic gives two valid solutions, then failing to reject one using a condition in the question, such as "the circle lies in the first quadrant".
- Not sketching. A quick sketch with the centre, radius and given line catches most errors and helps you decide which root to reject.
- Leaving coordinates as decimals when exact surd form is expected.
A Simple Practice Routine
Work through past year papers by question type rather than by year. Do ten "centre and radius" parts in a row, then ten tangent parts, then ten discriminant parts. Pattern recognition is what makes you fast in the exam hall, and speed on circles buys you time for the harder trigonometry or calculus questions later in the paper.
If circle questions still feel slow, our tutors at Dojo recently sat these exact papers and can walk you through the working line by line. Start with a free trial class on WhatsApp and bring along the question that has been bugging you.