Circle questions in Paper 1 combine coordinate geometry with algebra, and sometimes with calculus or sector areas. Marks are most often lost in finding the centre and radius, in sign errors, and in confusing the radius with the diameter or with the radius squared. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Radius, diameter and radius squared. Students found the distance between two points but did not realise that it was the diameter. Putting the radius, rather than the radius squared, on the right-hand side of the equation is a common slip, and so is an arithmetic error in r². One report notes that an answer left as the square root of 65 was not accepted for the radius squared.
- Signs in the centre. Reading the centre from (x − a)² + (y − b)² = r² gives (a, b), not (−a, −b). Reports mention both sign errors in the centre and sign errors when finding the gradient of a radius.
- The general form. Students who used x² + y² + 2gx + 2fy + c = 0 often did less well than those who used the standard centre-radius form, because of mistakes with the negative signs and with the relation between c, g, f and r.
- Brackets and squares. Omitting to square a bracket, swapping x and y, and mistakes when expanding brackets were all recorded.
- Not finding all that is needed. Candidates did not substitute y = 0 to find where tangents meet the x-axis, did not find the y-coordinates of the intersection points, or assumed a triangle was isosceles.
- Tangent and perpendicular questions. Students used the wrong gradient for the tangent (not the negative reciprocal of the radius's gradient), or found only one condition when two were needed. To show that a line is a tangent at a point, showing the point is on the circle and that the radius is perpendicular to the line are both needed.
- Using the discriminant wrongly. Reports mention a correct quadratic that was then not solved with the discriminant, wrong coefficients in the discriminant, or substituting x = 0.
- Combined questions. In questions with sectors, triangles or integration, students used half the required angle, used the wrong limits, or omitted a part of the area.
What examiners advise
- Sketch the circle, centre, tangent and radius. Several reports say that a diagram would have avoided wasted effort.
- Choose the standard form unless the question gives the general form. It makes the centre and radius easier to read.
- Show enough evidence. Say how you found the centre and radius. Insufficient evidence for the centre and radius cost marks in some reports.
- To find the intersection of a line and a circle, substitute for one variable and solve the resulting quadratic. Use the discriminant to decide on tangency.
- Remember what the question asks for, such as an exact value, the number of significant figures, or an equation.
Habits that help
- Write the centre and the radius (not radius squared) on separate lines before writing the equation.
- After finding the equation, substitute one given point to test it.
- Keep surds exact in a circle question unless the question asks for a decimal.
Questions students ask
How do I read the centre from (x − a)² + (y − b)² = r²?
The centre is (a, b), not (−a, −b). Sign errors in the centre are recorded in examiner reports, along with sign errors when finding the gradient of a radius.
What goes on the right-hand side of the equation of a circle?
The radius squared, not the radius. Putting r there is a common slip, and so is an arithmetic error in r². One report notes that an answer left as the square root of 65 was not accepted for the radius squared.
Is it better to use the standard form or the general form?
Examiners say the standard centre-radius form is easier unless the question gives the general form. Students who used x² + y² + 2gx + 2fy + c = 0 often did less well because of the negative signs and the relation between c, g, f and r.
What do I need to show that a line is a tangent to a circle?
Both conditions: that the point is on the circle, and that the radius to that point is perpendicular to the line. You can also substitute the line into the circle and use the discriminant. Students lose marks by finding only one condition or by using the wrong tangent gradient, which is the negative reciprocal of the radius's gradient.
How do I find where a line meets a circle?
Substitute for one variable and solve the resulting quadratic. The discriminant tells you whether the line is a tangent. Reports mention students who forgot to find the y-coordinates after finding x.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.