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

The relative geometric position of a straight line ax + by + k = 0 with respect to a circle (x - a)² + (y - b)² = r² is determined either by the perpendicular distance d from the centre to the line or by substituting the linear equation into the circle equation to form a quadratic. If d < r (or discriminant Δ > 0), the line is a secant intersecting the circle at two distinct real points. If d = r (or Δ = 0), the line is a tangent touching the circle at exactly one point. If d > r (or Δ < 0), the line lies entirely outside the circle with no real intersection.

Check before you answer (A/L Pure Maths Paper 1)

1. Perpendicular Distance Method: Calculating d = |a x₀ + b y₀ + k| / √(a² + b²) from centre (x₀, y₀) and comparing rigorously against r.
2. Condition of Tangency: Deriving and applying the tangency condition c² = a²(1 + m²) for the line y = mx + c and circle x² + y² = a².
3. Length of Intercepted Chord: Evaluating chord length 2√(r² - d²) using Pythagoras' theorem on the perpendicular distance from the centre.