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4MA1/2HR Jun 2020, Question 26: Chord Properties

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 5 marks

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The question, in short

Use the intersecting secants from external point C through the circle to find AB in the form p√5 + q, stating integer p and q.

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

Solution · 5 marks

  1. 1(2 + √5) × AC = (2√5) × (2√5 + 4 + √5)
  2. 2(2 + √5) × AC = (2√5) × (3√5 + 4)
  3. 3(2 + √5) × AC = 2√5 × 3√5 + 2√5 × 4
  4. 4(2 + √5) × AC = 6×5 + 8√5
  5. 5(2 + √5) × AC = 30 + 8√5
  6. 6AC = (30 + 8√5) / (2 + √5)
  7. 7Multiply numerator and denominator by (2 − √5)
  8. 8AC = (30 + 8√5)(2 − √5) / ((2 + √5)(2 − √5)
  9. 9Denominator: 4 − 5 = −1
  10. 10Numerator: 30×2 − 30×√5 + 8√5×2 − 8√5×√5
  11. 11Numerator: 60 − 30√5 + 16√5 − 40
  12. 12Numerator: 20 − 14√5
  13. 13AC = (20 − 14√5) / (−1) = −20 + 14√5
  14. 14AB = AC − BC = (−20 + 14√5) − (2 + √5)
  15. 15AB = −20 + 14√5 − 2 - √5
  16. 16AB = −22 + 13√5
  17. 17AB = 13√5 − 22

Answer13√5 − 22

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