This module covers how coordinate geometry and algebra meet on a circle: reading the centre and radius from an equation, finding where a line cuts a circle, drawing a tangent, finding a parameter that makes a line touch a circle, and writing an equation from a geometric description. One idea runs through all of it: a circle is every point at a fixed distance r from its centre, so every method starts from that distance.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content and notation in your exam year, because the boundaries between topics can shift. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to complete the square, solve quadratics and use the discriminant, and find a line through a point with a given gradient. If any of these feel slow, revise quadratic structure and discriminants and straight lines and linearisation first. Simultaneous equations from simultaneous linear and nonlinear models are the engine behind most circle questions.
An orienting example
The circle C has equation x² + y² − 4x − 6y − 12 = 0. Find its centre and radius, then find the tangent at the point P(5, 7).
Step 1, complete the square: (x − 2)² − 4 + (y − 3)² − 9 − 12 = 0, so (x − 2)² + (y − 3)² = 25.
Step 2, read off: centre (2, 3) and radius 5.
Step 3, check P is on the circle: (5 − 2)² + (7 − 3)² = 9 + 16 = 25. ✓
Step 4, radius gradient: from (2, 3) to (5, 7) the gradient is 4/3.
Step 5, tangent gradient: the tangent is perpendicular to the radius, so its gradient is −3/4.
Step 6, equation: y − 7 = −3/4 (x − 5), which gives 4y − 28 = −3x + 15, so 3x + 4y = 43.
Check: the distance from (2, 3) to the line is |6 + 12 − 43| ÷ 5 = 25 ÷ 5 = 5, equal to the radius. ✓
Two separate skills did the work here: completing the square and using perpendicular gradients. The lessons below teach each one on its own before they are combined.
In which order should you study it?
- Recover centre and radius by completing the square: the starting point for every other skill, since you cannot use a circle you cannot read.
- Find intersections of a line and circle: substitution turns the question into a quadratic, and the discriminant counts the points.
- Use a tangent perpendicular to the radius: brings in gradients and the fact that a tangent meets the radius at a right angle.
- Solve a parameter condition for tangency: joins lessons two and three, asking for the value of k that gives exactly one point.
- Translate a geometric condition into a circle equation: works in the other direction, building an equation from a diameter, a touching axis or equal distances.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Taking the centre with the wrong signs, such as (−3, 2) for (x − 3)² + (y + 2)² = 25.
- Giving r² as the radius, so a circle with r² = 25 is drawn with radius 25.
- Expanding (x + 2)² as x² + 4 and losing the middle term when substituting a line.
- Using the radius gradient as the tangent gradient, forgetting to flip and negate.
- Setting the discriminant greater than zero when tangency needs it equal to zero.
- Stopping at one value of k when the condition has a positive and a negative solution.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper first, and draw a rough sketch with the centre and radius marked before you calculate. Then open the worked answer and compare method as well as final value. Use the non-calculator working trainer to rehearse exact arithmetic with surds and fractions, and the quadratic structure explorer to check the discriminant of a quadratic that appears after substitution.
When you get something wrong, read the routing table at the end of the practice set and return to the lesson it names. The line and circle intersection explorer lets you slide a line across a circle and watch the number of intersections change. Keep a record in the mistake log and retest queue, and retry a fresh question a few days later.
If your progress stalls on the same habit, a teacher can look at your written solutions in online one-to-one Additional Mathematics tuition.