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0606/12 May/Jun 2025, Question 3: Circles Meeting Lines, and Tangents

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 8 marks

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The question, in short

For a circle with centre C given by x² + y² − 10x − 4y + 24 = 0, work with the line y = 2x − 3 and the point P where it touches the circle.

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

Part (a) · 3 marks

Demonstrate that y = 2x − 3 meets the circle at exactly one point.

  1. 1Substitute y = 2x − 3: x² + (2x − 3)² − 10x − 4(2x − 3) + 24 = 0
  2. 25x² − 30x + 45 = 0, i.e. x² − 6x + 9 = 0
  3. 3Discriminant = (−30)² − 4(5)(45) = 900 − 900 = 0
  4. 4One repeated root, so the line touches the circle: it is a tangent

AnswerSubstituting gives 5x² − 30x + 45 = 0, discriminant = 0 (one repeated root), so the line is a tangent (shown)

Part (b) · 2 marks

Using P as the contact point of the tangent, determine its coordinates.

  1. 1x² − 6x + 9 = 0 gives (x − 3)² = 0, so x = 3
  2. 2y = 2(3) − 3 = 3

Answer(3, 3)

Part (c) · 3 marks

Determine the equation of a circle centred at P that goes through the origin.

  1. 1Radius = distance from (3, 3) to (0, 0) = √(9 + 9) = 3√2
  2. 2r² = 18
  3. 3(x − 3)² + (y − 3)² = 18

Answer(x − 3)² + (y − 3)² = 18

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Also searched as: 0606/12/M/J/25 · 0606_s25_qp_12 · 0606 s25 12 · question 3.
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