Coordinate geometry in A Level Math Paper 1: common mistakes and examiner advice

Coordinate geometry should be a reliable source of marks, and the reports often say so. The mistakes are rarely deep. They are slips in gradients, midpoints and perpendicular lines, and they cost marks because they happen early in a multi-step question. 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

• Using the wrong point for the perpendicular bisector. The line goes through the midpoint of the segment, not through one of the end points. Using A or B instead of the midpoint is listed in the reports as one of the most frequent errors.
• Adding and subtracting mixed up. The midpoint comes from adding coordinates (then halving). The gradient comes from subtracting. Some students subtracted when finding the midpoint, or failed to take the difference in y over the difference in x.
• The wrong gradient. Students used the gradient of the line joining the two points when the perpendicular gradient was needed. Others inverted the gradient but forgot the sign change, or swapped x and y in the equation of a line.
• Showing a right angle incorrectly. To show that two lines are perpendicular, some students set the two gradient expressions equal to each other, or multiplied one expression by −1 and then equated them, instead of setting the product of the gradients equal to −1.
• Assuming things from the diagram. Reports mention students who assumed AD = CB, assumed that a point lay on a perpendicular bisector, or assumed that a line was horizontal. Diagrams are often not drawn to scale.
• Using the wrong point as the centre. When a centre had to be found, students used one of the given points or the midpoint of a side.
• Substituting the wrong coordinate. When finding where a line meets an axis, some students put x = 0 when they needed y = 0.
• Not using exact values. When an exact area was asked for, using decimal angles from a calculator introduced rounding errors.
• Algebra under pressure. Statements such as (k − 5)² = k² + 25 appear in many scripts. Sign slips in coordinates and in equations of lines are common.
• Confusing which formula is for which quantity. Length of AB and radius, or radius and radius squared. Answers in the form (√65) were not accepted for radius squared.

What examiners advise

• Draw a diagram. Reports say that students who drew one were invariably more successful, and that this should always be encouraged for coordinate geometry.
• Use all the information. Where a question gives two pieces of information, use both. Students who used only one could not solve their equations.
• Do not state a result without a method when finding a second point, or when solving a quadratic that arises.
• Cancel unknowns in gradients. If a gradient has the same unknown in numerator and denominator, simplify it before writing the final equation.
• Name what you are finding. Is it a gradient, a length, a squared length, a coordinate?

Habits that help

• Write the formula you are using before you substitute into it.
• Work out the midpoint and gradient on separate lines.
• After finding a point, check that it fits the diagram and the equations you were given.

Questions students ask

Which point does the perpendicular bisector go through?

The midpoint of the segment, not one of its end points. Examiner reports list using A or B instead of the midpoint as one of the most frequent errors.

How do I show that two lines are perpendicular?

Show that the product of the gradients is −1. Some students instead set the two gradient expressions equal to each other, or multiplied one by −1 and then equated them, and lost the marks.

Do I add or subtract coordinates for a midpoint and a gradient?

The midpoint comes from adding the coordinates and halving. The gradient comes from subtracting: the difference in y over the difference in x. Mixing these up is a recorded mistake, so write the formula before you substitute.

Can I trust the diagram in a coordinate geometry question?

No. Diagrams are often not drawn to scale. Reports mention students who assumed that two lengths were equal, that a point lay on a perpendicular bisector or that a line was horizontal. Use only what the question tells you, and draw your own sketch.

Why do examiners keep telling students to draw a diagram?

The reports say that students who drew one were invariably more successful. A sketch also helps you check that a point you found fits the diagram and the equations you were given.