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WMA12/01 Jun 2024, Question 7: Circles Meeting Lines, and Tangents

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 6 marks

The question, in short

For circle C₁ given by x² + y² + 8x − 10y = 29, determine its centre and exact radius; then show that C₁ and C₂, where (x − 5)² + (y + 8)² = 52, do not meet or touch.

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

Part (ai) · 2 marks

Work out the centre coordinates of C₁ from its expanded equation.

  1. 1(x + 4)² − 16 + (y − 5)² − 25 = 29
  2. 2(x + 4)² + (y − 5)² = 70
  3. 3Centre (−4, 5)

Answer(−4, 5)

Part (aii) · 1 mark

Calculate the radius of C₁ as an exact value.

  1. 1(x + 4)² + (y − 5)² = 70
  2. 2Radius = √70

Answer√70

Part (b) · 3 marks

Using detailed reasoning, demonstrate that the two circles neither intersect nor are tangent.

  1. 1Centre of C₂ is (5, −8), radius √52
  2. 2Distance between centres = √((5 − (−4))² + (−8 − 5)²) = √(81 + 169) = √250 ≈ 15.81
  3. 3Sum of radii = √70 + √52 ≈ 8.367 + 7.211 = 15.58
  4. 415.81 > 15.58, so the circles are separate: they neither touch nor intersect

AnswerShown: distance between centres = √250 ≈ 15.81, sum of radii = √70 + √52 ≈ 15.58; the distance is greater than the sum of the radii, so the circles neither touch nor intersect

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Also searched as: WMA12 June 2024 · WMA12/01 · wma12 jun24 · IAS Maths Pure Mathematics 2 June 2024 · question 7.
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