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4PM1/01 Nov 2023, Question 4: The Point Dividing a Line in a Ratio

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

Points A(12, 14) and B(q, 2) lie on the line 3y − 2x − p = 0. Find p and q, then find the equation of the line L perpendicular to AB through point X on AB with AX : XB = 1 : 2.

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Part (a) · 3 marks

Use the fact that A and B lie on 3y − 2x − p = 0 to find the constants p and q.

  1. 1A on the line: 3(14) − 2(12) − p = 0, so p = 18
  2. 2B on the line: 3(2) − 2q − 18 = 0
  3. 3q = −6

Answerp = 18, q = −6

Part (b) · 6 marks

Find the coordinates of X dividing AB in ratio 1 : 2, then find the equation of the perpendicular line L through X in the form ax + by + c = 0.

  1. 1X = A + (1/3)(B − A) = ((1(−6) + 2(12))/3, (1(2) + 2(14))/3) = (6, 10)
  2. 2Gradient of AB = 2/3
  3. 3Gradient of L = −3/2
  4. 4y − 10 = −(3/2)(x − 6)
  5. 53x + 2y − 38 = 0

Answer3x + 2y − 38 = 0

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