Circle Concepts and Equations

Overview

This lecture covers all essential concepts, formulas, and problem types from the chapter on circles, including equations, properties, tangents, chords, and mutual positions, aimed at preparing for college and HSC exams.

Circle Equation Basics

• The general equation of a circle is a quadratic in x and y with equal, nonzero coefficients for x² and y², and no xy term.
• Standard form: x² + y² + 2Gx + 2Fy + C = 0, where center is (-G, -F) and radius is √(G² + F² - C).
• Any quadratic equation represents a circle if coefficients of x² and y² are equal (not zero) and no xy term exists.
• To convert circle equations to standard form, divide all terms by the coefficient of x² or y² if it's not 1.

Key Properties and Problem Types

• Center: (-G, -F); Radius: √(G² + F² - C).
• If radius = 0, it’s a point circle; if G² + F² - C < 0, the circle is imaginary.
• To find the equation given center (h, k) and radius r: (x - h)² + (y - k)² = r².
• The equation of a circle through three points is found by solving the system for G, F, and C.
• The equation for a circle with diameter endpoints (x₁, y₁), (x₂, y₂): (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0.

Circle and Axes Interactions

• Circle touches x-axis if G² = C; touches y-axis if F² = C.
• If it touches both axes, G² = F² = C; the point of contact coordinates are (–G, 0) and (0, –F).
• Tangent at a point (x₁, y₁) on the circle: xx₁ + yy₁ + G(x + x₁) + F(y + y₁) + C = 0.

Tangents and Chords

• A straight line y = mx + c is tangent to the circle if the perpendicular distance from the center equals the radius.
• Tangent from external point (x₁, y₁): length = √[S₁], where S₁ is the value when (x₁, y₁) substituted into the circle’s equation.
• The chord of contact from external point: replace (x, y) with ((x + x₁)/2, (y + y₁)/2) in the circle's equation.

Position of Points and Circles

• Substitute (x₁, y₁) into the circle's equation:
  • If equals 0: point is on the circle.
  • If > 0: point lies outside.
  • If < 0: point lies inside.
• Distance between circle centers C₁C₂ compared to radii (r₁, r₂) determines if circles are separate, touching externally/internally, or intersecting.

Mutual Position of Circles

• C₁C₂ > r₁ + r₂: circles are separate.
• C₁C₂ = r₁ + r₂: circles touch externally.
• |r₁ - r₂| < C₁C₂ < r₁ + r₂: circles intersect at two points.
• C₁C₂ = |r₁ - r₂|: circles touch internally.
• C₁C₂ < |r₁ - r₂|: one circle lies completely inside the other.

Special Circle Forms & Coordinates

• Parametric equations: x = h + r cosθ, y = k + r sinθ.
• Polar form: r² + 2Gr cosθ + 2Fr sinθ + C = 0; center at (–G, –F), radius √(G² + F² – C).

Key Terms & Definitions

• Circle standard equation — x² + y² + 2Gx + 2Fy + C = 0.
• Radius — Distance from center to any point on the circle; √(G² + F² – C).
• Tangent — A line touching the circle at exactly one point.
• Chord of contact — Line joining the points of tangency from an external point.
• Point circle — Circle with zero radius.
• Imaginary circle — Circle with a complex-valued radius.

Action Items / Next Steps

• Review circle chapter in textbook prior to revising these notes.
• Practice converting general quadratic to standard form and solving for center/radius.
• Solve example problems involving tangents, chords, and circle intersection types.
• Memorize formulas, especially for tangents, circle equations, and distance conditions.
• Attempt all relevant MCQs and written problems from your exam board materials.