Coordinate geometry questions combine midpoints, gradients, perpendicular lines, distances, areas and, for the circle, the equation of a circle, tangents and intersections. The reports show that the method is usually known and that the marks are lost on small things: which point to use, what form the answer takes, and which steps to show. This guide collects what Cambridge examiners say students get wrong, based on recent reports, for both the straight line and the circle.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks: the straight line
- The perpendicular bisector. The main error: using the coordinates of A or B instead of the midpoint. Reports say a "bisector" built from one of the given points earns no marks. Others did not find the midpoint at all, used the gradient of AB instead of the perpendicular gradient, or did not use the perpendicular gradient. Arithmetic slips in the midpoint or gradient were also common, and the reports say that candidates who showed the method for the midpoint and gradient still earned method marks.
- Leaving out y. Offering "equation = −3x + 25", or writing "BD = ...", without y in the answer.
- The required form. Not giving the final answer in the form ax + by + c = 0 with integer coefficients, or giving a bisector with non-integer coefficients.
- Exact values. When a coordinate can be written exactly (a fraction or a terminating decimal, such as −1.9375), rounded decimals are not acceptable.
- Starting with wrong points. Sign errors when finding P and Q, which meant that later work was based on wrong points. The reports say this was not accepted.
- The distance formula. An incorrect formula for the length, or errors in using it.
- Assuming a right angle. In one question, students assumed that the triangle was right-angled at A or B, which often gave an answer close to the correct one.
- Missing a required element. In a question with three required elements, the proof of perpendicularity was the one most commonly omitted. With the shoelace (determinant) method, a few omitted the ½ or the repeated point.
- Not using the mark allocation. The reports say a two-mark question signals a straightforward method, and that mark allocation and space guide how much work is expected.
Mistakes that cost marks: the circle
- The equation. Sign errors, or equating to 5 rather than 25. Some students could not recall the standard or general equation of a circle, which is in the list of formulas on page 2. Some used a wrong formula for the distance between two points, such as (x − h)² − (y − k)². Others did not state r² as an integer.
- Centre and radius. Assuming the centre was at (5, 5) or (6, 6), labelling the diameter as the radius, and unclear working, with the centre or radius not clearly identified before they were inserted into the equation. Reports note that the centre should be found by recognising the x-coordinate lies halfway between the lines x = 0 and x = 4.
- Comparing with the standard form. The square root has to be applied to get r from r².
- Tangents. Calculating the gradient of the radius and not taking the negative reciprocal for the tangent. Others did not give the tangent equation in the required form.
- Two circles. Many did not subtract the equations to find the common chord. Others treated the problem as an identity and solved (x + 1)² = (x − 6)² and (y − 2)² = (y − 3)² separately, wrongly cancelled the −20 on one side with the +20 on the other, or made a slip when rewriting in standard form and omitted the negative square root. Too many started finding perpendicular bisectors and then trying to find points of intersection, which was not efficient and often broke down.
- Algebra and method. Slips when using the discriminant, using a calculator to solve an incorrect quadratic in k, using only the equation of C₂ and not the information given for C₁, and omitting the elimination step or writing the equation in a solvable form, which suggested over-use of the calculator.
What examiners advise
- Draw a simple sketch. The reports say a helpful sketch was made by a few, and that a sketch may have helped many.
- Use the midpoint for the perpendicular bisector, and write every step.
- Check the form of the final answer against the question before moving on, including integer coefficients and the "y".
- Keep exact values when they can be written exactly.
- Refer to the list of formulas on page 2 for the equation of a circle.
- Use "hence" as a guide to the method.
- Set out each step clearly, because for full marks no step should be assumed.
Habits that help
- Write the midpoint and the gradient on separate lines before the equation.
- To check an equation of a line, substitute both given points into it.
- For a circle, write the centre and r² as soon as you have them, then insert them into the equation.
Questions students ask
Which point does a perpendicular bisector pass through?
The midpoint. Using A or B instead is the main error. Reports say a "bisector" built from one of the given points earns no marks.
What form should the equation of a line be in?
The form the question asks for, usually ax + by + c = 0 with integer coefficients, and do not leave out the y. Offering "equation = −3x + 25" loses marks.
How do I find the equation of a circle?
Write the centre and r² as soon as you have them, then insert into the equation. Labelling the diameter as the radius, and assuming a centre such as (5, 5), are recorded errors. The standard and general equations are in the list of formulas on page 2.
How do I find the equation of a tangent to a circle?
Find the gradient of the radius, then take the negative reciprocal for the tangent. Give the equation in the required form.
How do I find the common chord of two circles?
Subtract the equations. Many candidates did not, and instead solved separate equations in x and y.
Should I keep exact values?
Yes, when a coordinate can be written exactly as a fraction or terminating decimal, such as −1.9375. Rounded decimals are not acceptable.
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