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WMA11/01 Jan 2024, Question 5: Perpendiculars: Feet and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Given line l₁ through P(−2, 9) and Q(10, 6), find its equation, the point R where it meets a perpendicular line through O, and the exact area of triangle OPQ.

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

Part (a) · 3 marks

Find the equation of l₁ in the form y = mx + c, determining constants m and c.

  1. 1m = (9 − 6)/(−2 − 10) = 3/(−12) = −1/4
  2. 2y − 9 = −¼(x + 2)
  3. 3y = −¼x + 17/2

Answery = −¼x + 17/2

Part (b) · 4 marks

Line l₂ through O is perpendicular to l₁; find the coordinates of their intersection R.

  1. 1Perpendicular gradient = 4, through O: l₂ is y = 4x
  2. 24x = −¼x + 17/2
  3. 3(17/4)x = 17/2 ⇒ x = 2
  4. 4y = 4 × 2 = 8, so R(2, 8)

AnswerR(2, 8)

Part (c) · 3 marks

Find the exact area of triangle OPQ.

  1. 1OR is perpendicular to PQ, so OR is the height of triangle OPQ
  2. 2PQ = √(12² + 3²) = √153 = 3√17
  3. 3OR = √(2² + 8²) = √68 = 2√17
  4. 4Area = ½ × 3√17 × 2√17 = ½ × 6 × 17 = 51

Answer51

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