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Circle Geometry Guide

What should I know about Standard and General Form?

Standard form: (x − a)² + (y − b)² = r². Where (a, b) is centre, r is radius. Example: (x−3)² + (y+2)² = 25. Centre (3, −2), radius 5. General form: x² + y² + Dx + Ey + F = 0. Convert by completing the square. Example: x² + y² − 4x + 6y + 9 = 0. (x−2)² − 4 + (y+3)² − 9 + 9 = 0. (x−2)² + (y+3)² = 4. Centre (2, −3), radius 2. Relationships: D = −2a. E = −2b. F = a² + b² − r². The general form requires the coefficients of x² and y² to be equal (so each can complete to a square) — otherwise it's not

What's the key thing to understand about Core Formulas?

Diameter d = 2r. Circumference C = 2πr = πd. Area A = πr². Where π ≈ 3.14159. From C find r: r = C / (2π). Example: C = 31.4 → r = 31.4 / 6.28 = 5. From A find r: r = √(A/π). Example: A = 50 → r = √(50/π) = √15.92 = 3.99. Sector of angle θ (degrees): arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². Sector of angle θ (radians): arc length = rθ. Sector area = ½r²θ. Segment of a chord: area = ½r²(θ − sin θ) where θ is in radians.

What do I need to know about Equations of Tangent Lines?

Tangent to circle x² + y² = r² at point (x₁, y₁): xx₁ + yy₁ = r². Tangent to (x−a)² + (y−b)² = r² at (x₁, y₁): (x−a)(x₁−a) + (y−b)(y₁−b) = r². Tangent length from external point: √(distance² − r²). Two circles: tangent if distance between centres equals sum (externally tangent) or absolute difference (internally tangent) of radii. Intersecting if difference < distance < sum. Power of a point P with respect to circle: |PC|² − r². Where C = centre. The power equals (tangent length)² for external points, a relationship used in various geometric proofs and constructions involving circles and tangent lines.

What should I know about Applications?

Engineering: gear ratios depend on circle circumference. Architecture: arches, domes use circular geometry. Astronomy: orbital paths approximated as circles (more accurately ellipses, but circles are first approximation). Navigation: great circles — shortest paths on Earth's surface. Statistics: circle and pi appear in normal distribution (∫e^(-x²)dx). Physics: angular velocity ω = 2π/T (period). Centripetal acceleration a = v²/r. Fourier analysis: circles in the complex plane underlie all of signal processing, since sine and cosine waves are fundamentally projections of circular motion.