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4PM1/01R Jan 2020, Question 9: Length of a Line Segment

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 15 marks

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The question, in short

Line l₁: y + 2x − 4 = 0 passes through P(a, 6) and Q(3, b). Find a and b, then use perpendicular line l₂ through P with PR = 6√5 to find R, area PQR, and circle centre.

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

Part (a) · 2 marks

Substitute the coordinates of P and Q into the equation of l₁ to find a and b.

  1. 16 + 2a − 4 = 0, so a = −1
  2. 2b + 6 − 4 = 0, so b = −2

Answera = −1, b = −2

Part (b) · 8 marks

Find the gradient of l₂, then the two points R on l₂ a distance 6√5 from P, giving e and f.

  1. 1Gradient of l₁ = −2, so gradient of l₂ = 1/2
  2. 2l₂: y − 6 = ½(x + 1), so x = 2y − 13
  3. 3PR² = 180: (x + 1)² + (y − 6)² = 180
  4. 4(2y − 12)² + (y − 6)² = 180, so 5(y − 6)² = 180
  5. 5y − 6 = ±6: y = 0 or y = 12
  6. 6x = −13 or x = 11: R = (−13, 0) or (11, 12)

Answere = −13, f = 0 and e = 11, f = 12

Part (c) · 3 marks

Using the condition e < 0, select R and calculate the area of triangle PQR.

  1. 1PQ = √((3 + 1)² + (−2 − 6)²) = √80 = 4√5
  2. 2Angle QPR = 90°, PR = 6√5
  3. 3Area = ½ × 4√5 × 6√5 = 60

Answer60

Part (d) · 2 marks

Find the centre of the circle through P, Q and R.

  1. 1Angle RPQ = 90°, so RQ is a diameter
  2. 2Centre = midpoint of RQ = ((−13 + 3)/2, (0 − 2)/2) = (−5, −1)

Answer(−5, −1)

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