Position of a straight line relative to a circle — A/L Combined Mathematics

How do I practise Position of a straight line relative to a circle?

1 question with a worked solution

Check before you answer: Perpendicular Distance Method: Calculating d = |a x₀ + b y₀ + k| / √(a² + b²) from centre (x₀, y₀) and comparing rigorously against r.

1. What is the condition on the constant c for the straight line y = 2x + c to be a tangent to the circle x² + y² = 5?
   1. A) c = ±√5
   2. B) c = ±10
   3. C) c = ±25
   4. D) c = ±5
   Check my answer

   Correct answer: D) c = ±5

   The circle x² + y² = r² has radius r = √5 and center at (0, 0). The line is 2x - y + c = 0. For tangency, the perpendicular distance from center (0, 0) to line must equal radius r: |c| / √(2² + (-1)²) = √5 => |c| / √5 = √5 => |c| = 5 => c = ±5.

   Show worked solution

   Step 1: Condition of tangency: d = r => |c| / √(m² + 1) = r. Step 2: Here m = 2, r = √5 => |c| / √(4 + 1) = √5 => |c| / √5 = √5 => c = ±5. Answer: c = ±5.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Position of a straight line relative to a circle fit in the syllabus?

Year 1 · Term 3