Coordinate Geometry
Analyse lines and circles using equations, gradients, intersections, chords and tangents.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Choose a line equation that exposes the given point, gradient or coefficients.
Exam cue: Sketch the relative position before solving.
Concept 2
Move between centre-radius and expanded circle forms.
Exam cue: Complete the square to expose a circle's centre and radius.
Concept 3
Combine algebra with perpendicular, chord, tangent and circle-angle properties.
Exam cue: Interpret a repeated intersection root as tangency.
Risk pitfalls and guardrails
Using the same gradient for perpendicular lines.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Losing a solution when eliminating a variable.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Treating every line-circle pair as having two real intersections.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Point-gradient
y − y₁ = m(x − x₁) uses one point and one gradient.
Perpendicular
For non-vertical lines, m₁m₂ = −1.
Circle centre
(x − a)² + (y − b)² = r² has centre (a,b).
Radius to tangent
The radius is perpendicular at the point of contact.
Chord bisector
The perpendicular from the centre bisects a chord.
Intersection count
The resulting equation reveals zero, one or two contacts.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
A line through (2,−1) is perpendicular to 3x−2y=7. Where does it meet the y-axis?
The midpoint of A(−3,5) and B is (2,−1). What is the squared distance AB²?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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