Coordinate Geometry of The Circle (Copy)

8 Coordinate Geometry of the Circle – Cheat Sheet

Equation of a Circle

• Standard form: (x – a)² + (y – b)² = r²
  • Centre = (a, b)
  • Radius = r
• General form: x² + y² + 2gx + 2fy + c = 0
  • Centre = (–g, –f)
  • Radius = √(g² + f² – c)

Intersection of a Straight Line & Circle

1. Solve simultaneous equations:
   • Circle: (x – a)² + (y – b)² = r²
   • Line: y = mx + c or ax + by + d = 0
2. Discriminant method:
   • If Δ > 0 → 2 intersection points (chord)
   • If Δ = 0 → 1 point (tangent)
   • If Δ < 0 → no intersection

Tangent to a Circle

• Condition: Distance from centre to line = radius
  • For line ax + by + c = 0 and centre (h, k):
    Distance = |a·h + b·k + c| / √(a² + b²) = r
• Equation of tangent at point (x₁, y₁) on circle:
  (x₁ – a)(x – a) + (y₁ – b)(y – b) = r²

Intersection of Two Circles

1. Circle 1: (x – a₁)² + (y – b₁)² = r₁²
2. Circle 2: (x – a₂)² + (y – b₂)² = r₂²
3. Subtract equations → linear equation for common chord
4. Solve with one circle equation to find intersection points

• Conditions:
  • Distance between centres < r₁ + r₂ and > |r₁ – r₂| → intersect at 2 points
  • Distance = r₁ + r₂ or = |r₁ – r₂| → touch (externally / internally)
  • Distance > r₁ + r₂ or < |r₁ – r₂| → no intersection