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4MA1/1H Jun 2023, Question 19: Vector Geometry

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 5 marks

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The question, in short

Triangle OAB with M midpoint of OA and ON:NB = 1:5; vectors OM = a and ON = b. Find vector AN and, using vectors, the ratio AP:PN where lines MB and NA meet at P.

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

Part (a) · 1 mark

Express the vector AN in terms of a and b.

  1. 1M is the midpoint of OA, so OA = 2 × OM = 2a
  2. 2AN = AO + ON
  3. 3AN = −2a + b

Answer−2a + b

Part (b) · 4 marks

Using a vector method, determine the ratio AP : PN.

  1. 1ON : NB = 1 : 5, so OB = 6 × ON = 6b
  2. 2MB = MO + OB = −a + 6b
  3. 3P lies on AN, so let AP = x(−2a + b), giving OP = OA + AP = 2a + x(−2a + b) = (2 − 2x)a + xb
  4. 4P also lies on MB, so let MP = y(−a + 6b), giving OP = OM + MP = a + y(−a + 6b) = (1 − y)a + 6yb
  5. 5Comparing the a terms: 2 − 2x = 1 − y
  6. 6Comparing the b terms: x = 6y
  7. 7Substituting: 2 − 12y = 1 − y, so 11y = 1, y = 1/11 and x = 6/11
  8. 8So AP = (6/11)AN and PN = AN − AP = (5/11)AN
  9. 9AP : PN = 6 : 5

Answer6 : 5

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