Coordinate geometry
A-level Maths (9MA0) · exam-style practice, examiner-report intelligence and the tools that drill it.
The topic on one screen
- Circle equation: (x - a)2 + (y - b)2 = r2. Given the expanded form, complete the square in x and y to extract centre and radius.
- Perpendicular gradients multiply to -1; parallel lines share a gradient. Most line questions are one of these two facts.
- A tangent to a circle is perpendicular to the radius at the point of contact — that's the whole method.
- Line meets circle: substitute and read the discriminant. Two roots = crosses, repeated root = tangent, no roots = misses.
- For centres/radii in terms of a constant k, the radius must be real and positive: r2 > 0 gives the condition on k.
- Distance between points: √[(x2-x1)2 + (y2-y1)2] — leave surds exact unless told otherwise.
Where students actually lose marks
In the circle-with-k question the centre and r2 must be EXTRACTED from the completed-square form — quoting r2 still buried inside the equation doesn't score, and ±√(…) for a radius is not allowed.
June 2023 Paper 1 mark scheme (Q10)
The scheme condones arithmetic slips inside the rearrangement but requires an attempted b2 - 4ac with at least one critical value for the intersection condition — a bare inequality with no discriminant work scores nothing.
June 2023 Paper 1 mark scheme (Q10)
Radius answers that then get 'simplified' further (e.g. divided by 2 after the square root) lose the mark — schemes do not apply isw when the extra step changes the value.
June 2023 Paper 1 mark scheme (Q10)
Try it — exam-style
A circle has equation x2 + y2 - 4kx + 6ky + 9 = 0, where k is a constant. Find, in terms of k, (i) the centre and (ii) the radius, and (iii) state the condition on k for the circle to exist.
The circle (x - 3)2 + (y + 2)2 = 25 passes through the point P(6, 2). Find the equation of the tangent to the circle at P.
Line L1 has equation y = 3x - 7. Line L2 passes through (6, 1) and is perpendicular to L1. Find the equation of L2.
A(1, 5) and B(7, -3) are the ends of a diameter of a circle. Find the equation of the circle.
Questions are written in the style of past Edexcel papers (source shown on each) — never copied from them.
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