📐 Core Concepts
① Standard Form of a Circle
A circle with center (h, k) and radius r:
(x − h)² + (y − k)² = r²
Special case — center at origin:
x² + y² = r²
② General Form & Completing the Square
General form: x² + y² + Dx + Ey + F = 0
Convert by completing the square:
Convert by completing the square:
(x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F
Center = (−D/2, −E/2), Radius² = (D/2)² + (E/2)² − F
③ Key Properties & Intersections
• Tangent line at point (x₁, y₁) on x²+y²=r²: x·x₁ + y·y₁ = r²
• Distance from center to line = radius (tangency condition)
• Two circles: subtract equations → find radical axis
• Circle & line: substitute → discriminant determines intersection type
• Distance from center to line = radius (tangency condition)
• Two circles: subtract equations → find radical axis
• Circle & line: substitute → discriminant determines intersection type
📝 Questions