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4PM1/02R Jun 2023, Question 8: The Point Dividing a Line in a Ratio

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 16 marks

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The question, in short

Given A(1, 5) and B(9, 9), work through line AB, a perpendicular line l through point X dividing AB in ratio 3:1, point C(6, p) on l, and parallelogram ABCD with negative x-coordinate for D.

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

Part (a) · 3 marks

Find the equation of line AB through (1, 5) and (9, 9) in the form ax + by + c = 0 with integer coefficients.

  1. 1Gradient of AB = (9 − 5)/(9 − 1) = 1/2
  2. 2y − 5 = (1/2)(x − 1)
  3. 32y − 10 = x − 1
  4. 4x − 2y + 9 = 0

Answerx − 2y + 9 = 0

Part (b) · 5 marks

Show that the perpendicular to AB through X, where AX : XB = 3:1, has equation y = −2x + 22.

  1. 1X = ((3 × 9 + 1)/4, (3 × 9 + 5)/4) = (7, 8)
  2. 2Perpendicular gradient = −1/(1/2) = −2
  3. 3y − 8 = −2(x − 7)
  4. 4y = −2x + 22

Answery = −2x + 22 shown

Part (c) · 1 mark

Given C(6, p) lies on line l, find the value of p.

  1. 1p = −2 × 6 + 22 = 10

Answerp = 10

Part (d) · 3 marks

For parallelogram ABCD with D having negative x-coordinate, find the coordinates of D.

  1. 1In parallelogram ABCD, CD = BA
  2. 2BA = (1 − 9, 5 − 9) = (−8, −4)
  3. 3D = C + BA = (6 − 8, 10 − 4)
  4. 4D = (−2, 6)

AnswerD(−2, 6)

Part (e) · 4 marks

Find the area of parallelogram ABCD.

  1. 1AB = √((9 − 1)² + (9 − 5)²) = √80
  2. 2CX is perpendicular to AB: CX = √((10 − 8)² + (6 − 7)²) = √5
  3. 3Area = base × height = √80 × √5
  4. 4= √400 = 20

Answer20

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