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WMA11/01 Jun 2024, Question 9: The Quadratic Formula and Completing the Square

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Given points A(4, 2) and B(15, 7) with line l₁ through them, find l₁'s equation, then use a horizontal line through A and BC = 5√5 to find C and the least area of triangle ABC.

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

Part (a) · 3 marks

Find an equation for the line through A(4, 2) and B(15, 7) in the form px + qy + r = 0 with integer coefficients p, q, r.

  1. 1Gradient = (7 − 2)/(15 − 4) = 5/11
  2. 2y − 2 = (5/11)(x − 4)
  3. 311y − 22 = 5x − 20
  4. 45x − 11y + 2 = 0

Answer5x − 11y + 2 = 0

Part (b) · 4 marks

Line l₂ through A is parallel to the x-axis. Point C on l₂ has BC = 5√5; find both possible coordinate pairs for C.

  1. 1l₂ is y = 2, so C = (x, 2)
  2. 2BC² = (15 − x)² + (7 − 2)² = (5√5)² = 125
  3. 3(15 − x)² = 100, so 15 − x = ±10
  4. 4x = 5 or x = 25: C = (5, 2) or (25, 2)

AnswerC = (5, 2) or C = (25, 2)

Part (c) · 2 marks

Using your points C, find the smallest possible area of triangle ABC.

  1. 1Base AC lies on y = 2; height = 7 − 2 = 5
  2. 2Smaller base: C = (5, 2), AC = 5 − 4 = 1
  3. 3Area = ½ × 1 × 5 = 5/2

AnswerMinimum area = 5/2 (= 2.5)

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