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0606/13 Oct/Nov 2025, Question 11: Circles Meeting Lines, and Tangents

Cambridge IGCSE Additional Mathematics 0606 · Paper 13 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

A circle has tangents x = 0, x = 4, y = 3 and y = −1; find its equation, then use the tangent y = 2x + a to derive a quadratic and find a.

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Part (a) · 3 marks

Find the equation of the circle whose tangents are the lines x = 0, x = 4, y = 3 and y = −1.

  1. 1The centre is midway between x = 0 and x = 4, and between y = −1 and y = 3: centre (2, 1)
  2. 2Radius = 4/2 = 2
  3. 3(x − 2)² + (y − 1)² = 4

Answer(x − 2)² + (y − 1)² = 4

Part (b) · 4 marks

Show that substituting y = 2x + a into the circle gives 5x² + 4(a − 2)x + (a − 1)² = 0, then find a exactly.

  1. 1(x − 2)² + (2x + a − 1)² = 4
  2. 2x² − 4x + 4 + 4x² + 4(a − 1)x + (a − 1)² = 4
  3. 35x² + 4(a − 2)x + (a − 1)² = 0
  4. 4Tangent so discriminant = 0: 16(a − 2)² − 20(a − 1)² = 0
  5. 5a² + 6a − 11 = 0
  6. 6a = −3 ± √20 = −3 ± 2√5

Answer5x² + 4(a − 2)x + (a − 1)² = 0 shown; a = −3 ± √20 (= −3 ± 2√5)

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