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0606/12 Oct/Nov 2025, Question 7: The Equation of a Circle

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 6 marks

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The question, in short

Points A(−2, 4), B(6, 10) and C(12, 2) are given. Find the gradients of AB, AC and BC, then use them to find the equation of the circle through A, B and C.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Calculate the gradient of each of the three lines AB, AC and BC joining the given points.

  1. 1AB: (10 − 4)/(6 − (−2)) = 6/8 = 3/4
  2. 2AC: (2 − 4)/(12 − (−2)) = −2/14 = −1/7
  3. 3BC: (2 − 10)/(12 − 6) = −8/6 = −4/3

AnswerGradient AB = 3/4, gradient AC = −1/7, gradient BC = −4/3

Part (b) · 4 marks

Using your gradients, determine the equation of the circle that passes through A, B and C.

  1. 1(3/4) × (−4/3) = −1, so AB ⊥ BC and angle ABC = 90°
  2. 2So AC is a diameter (angle in a semicircle)
  3. 3Centre = midpoint of AC = (5, 3)
  4. 4r² = (−2 − 5)² + (4 − 3)² = 49 + 1 = 50
  5. 5(x − 5)² + (y − 3)² = 50

Answer(x − 5)² + (y − 3)² = 50

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