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4PM1/02 Jun 2023, Question 8: Length of a Line Segment

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 17 marks

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The question, in short

Given points A(−6, 8) and B(12, 2), work through line AB, its length, a point X dividing AB in ratio 1:2, a perpendicular line L through X, and a circle with diameter AB.

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

Part (a) · 3 marks

Determine the equation of the line through A and B, giving integer coefficients a, b, c in ax + by + c = 0.

  1. 1Gradient of AB = (2 − 8)/(12 − (−6)) = −6/18 = −1/3
  2. 2y − 2 = −(1/3)(x − 12)
  3. 33y − 6 = −x + 12
  4. 4x + 3y − 18 = 0

Answerx + 3y − 18 = 0

Part (b) · 2 marks

Calculate the exact distance between A and B.

  1. 1AB = √(18² + (−6)²)
  2. 2= √360
  3. 3= 6√10

Answer6√10

Part (c) · 2 marks

Point X(m, n) on AB satisfies AX : XB = 1 : 2; determine m and n.

  1. 1X divides AB in the ratio 1 : 2
  2. 2m = (2 × (−6) + 1 × 12)/3 = 0
  3. 3n = (2 × 8 + 1 × 2)/3 = 6

Answerm = 0, n = 6

Part (d) · 7 marks

Line L through X is perpendicular to AB; C(p, q) on L has p > 0, q > 0 and lies on the circle with diameter AB. Find exact p and q.

  1. 1L is perpendicular to AB through X(0, 6): gradient 3, so q = 3p + 6
  2. 2C is on the circle with diameter AB, so angle ACB = 90°: CA ⊥ CB
  3. 3(q − 8)/(p + 6) × (q − 2)/(p − 12) = −1, so q² − 10q + 16 = −p² + 6p + 72
  4. 4Substitute q = 3p + 6: 10p² − 80 = 0
  5. 5p² = 8, p = √8 = 2√2 (p > 0)
  6. 6q = 3√8 + 6 = 6 + 6√2

Answer(i) p = √8 = 2√2 (ii) q = 6 + 6√2

Part (e) · 3 marks

Calculate the exact area of triangle ABC.

  1. 1CX = √((√8 − 0)² + (6 + 6√2 − 6)²) = √(8 + 72) = 4√5
  2. 2CX is perpendicular to AB
  3. 3Area = (1/2) × AB × CX = (1/2) × 6√10 × 4√5
  4. 4= 60√2

Answer60√2

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