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WMA12/01 Oct 2021, Question 6: Circles Meeting Lines, and Tangents

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

For circle C₁ given by x² + y² + 10x − 12y = k, find its centre and the possible values of k; then for circle C₂ through P(p, 0), Q(−2, 10), R(8, −14) with QR a diameter and p positive, find p.

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

Part (ia) · 2 marks

Find the coordinates of the centre of circle C₁ from its equation.

  1. 1(x + 5)² − 25 + (y − 6)² − 36 = k
  2. 2(x + 5)² + (y − 6)² = k + 61
  3. 3Centre (−5, 6)

Answer(−5, 6)

Part (ib) · 2 marks

State the range of possible values of the constant k.

  1. 1r² = k + 61
  2. 2The radius must be positive: k + 61 > 0
  3. 3k > −61

Answerk > −61

Part (ii) · 4 marks

Given QR is a diameter of C₂ and p is positive, find the exact value of p.

  1. 1Centre = midpoint of QR = ((−2 + 8)/2, (10 − 14)/2) = (3, −2)
  2. 2Radius = ½√((−10)² + 24²) = ½ × 26 = 13
  3. 3C₂: (x − 3)² + (y + 2)² = 169
  4. 4P(p, 0): (p − 3)² + 4 = 169 ⇒ (p − 3)² = 165
  5. 5p = 3 ± √165; p > 0, so p = 3 + √165

Answerp = 3 + √165

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Also searched as: WMA12 October 2021 · WMA12/01 · wma12 oct21 · IAS Maths Pure Mathematics 2 October 2021 · question 6.
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