4PM1/02 Jun 2022, Question 9: Perpendicular Lines
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 12 marks
اقرأ هذا الحل مشروحاً بالعربي ←
The question, in short
Two perpendicular lines through A(4, 7) with gradient m < 0 and B(4, k) meet at C with y-coordinate Y; show the given formula for Y, then use an isosceles triangle ABC to find m.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 7 marks
Show that the y-coordinate of the intersection point C equals (7 + m²k)/(m² + 1).
- 1L₁: y − 7 = m(x − 4)
- 2Gradient of L₂ = −1/m, so L₂: y − k = −(1/m)(x − 4)
- 3From L₁: x − 4 = (y − 7)/m
- 4Substitute: y − k = −(y − 7)/m²
- 5m²y − m²k = 7 − y ⇒ y(m² + 1) = 7 + m²k
- 6Y = (7 + m²k)/(m² + 1)
AnswerY = (7 + m²k)/(m² + 1) shown
Part (b) · 5 marks
Given triangle ABC is isosceles, find the value of m.
- 1Angle ACB = 90°, so the isosceles triangle has CA = CB and C lies level with the midpoint of AB
- 2Midpoint of AB = (4, (k + 7)/2), so (k + 7)/2 = (7 + m²k)/(m² + 1)
- 3(k + 7)(m² + 1) = 14 + 2m²k ⇒ m²(k − 7) = k − 7
- 4k ≠ 7 so m² = 1, m = ±1
- 5m < 0 so m = −1
Answerm = −1
Stuck on a step? Read the free lesson: Perpendicular Lines
Practise more perpendicular lines questions
Real 4PM1 past-paper questions, marked on your working like the exam, with a worked solution for every part.
Practise free on LemmaraMore past-paper questions on perpendicular lines
Also searched as: 4PM1 2 June 2022 · 4PM1/02 · 4pm1 jun22 2 · question 9.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.