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4PM1/01 Jan 2022, Question 10: Length of a Line Segment

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 17 marks

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The question, in short

Line L₁ has equation y − 2x − 6 = 0 and point P(2, a) lies on it. Find a, find the perpendicular line L₂ through P, then use points A, B, C to find m, n and the area of ACBP.

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

Part (a) · 1 mark

Substitute the coordinates of P into the equation of L₁ to find the value of a.

  1. 1a − 4 − 6 = 0, so a = 10

Answera = 10

Part (b) · 4 marks

Using the gradient of L₁ and point P, derive the equation of the perpendicular line L₂ and show it rearranges to x + 2y − 22 = 0.

  1. 1Gradient of L₁ = 2, so gradient of L₂ = −1/2
  2. 2y − 10 = −½(x − 2)
  3. 32y − 20 = −x + 2
  4. 4x + 2y − 22 = 0

Answerx + 2y − 22 = 0 shown

Part (c) · 9 marks

Find where L₁ and L₂ meet the x-axis to get A and B. With C(m, n), m > 0, n < 0, AC = 5√2 and gradient of BC = 1/4, find m and n.

  1. 1A = (−3, 0), B = (22, 0)
  2. 2AC² = 50: (m + 3)² + n² = 50
  3. 3Gradient BC: n/(m − 22) = 1/4, so m = 22 + 4n
  4. 4(25 + 4n)² + n² = 50, so 17n² + 200n + 575 = 0
  5. 5(n + 5)(17n + 115) = 0, so n = −5 or n = −115/17
  6. 6n = −5 gives m = 2 (n = −115/17 gives m < 0, rejected)

Answerm = 2, n = −5

Part (d) · 3 marks

Using the coordinates of A, C, B and P, calculate the area of quadrilateral ACBP.

  1. 1AB = 22 − (−3) = 25 along the x-axis
  2. 2Area APB = ½ × 25 × 10 = 125
  3. 3Area ABC = ½ × 25 × 5 = 125/2
  4. 4Area ACBP = 125 + 125/2 = 375/2

Answer375/2 (187.5)

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