Circles

Standard Forms of Equation of a Circle

General Form

• x² + y² + 2gx + 2fy + c = 0
• Centre: (−g, −f)
• Radius: √(g² + f² − c)

Centre–Radius Form

• (x − h)² + (y − k)² = a²
• Centre: (h, k), Radius: a
• If centre is origin: x² + y² = r²

Diameter Form

• If (x₁, y₁) and (x₂, y₂) are endpoints of diameter:
• (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0

Parametric Equations of a Circle

• For x² + y² = a²:
• x = a cosθ, y = a sinθ
• For (x − h)² + (y − k)² = a²:
• x = h + a cosθ, y = k + a sinθ
• For x² + y² + 2gx + 2fy + c = 0:
• x = −g + √(g² + f² − c) cosθ, y = −f + √(g² + f² − c) sinθ

Circle in Special Positions

Touching Coordinate Axes

• Touching both axes: (x ± a)² + (y ± a)² = a²
• Touching x-axis at (h, 0): (x − h)² + (y − a)² = a²
• Touching y-axis at (0, k): (x − a)² + (y − k)² = a²

Chord & Intercepts

• Length of chord at distance p from centre:
• Chord length = 2√(a² − p²)
• For x² + y² + 2gx + 2fy + c = 0:
• Intercept on x-axis: 2√(g² − c)
• Intercept on y-axis: 2√(f² − c)

Position of a Point w.r.t. a Circle

• For point (x₁, y₁), evaluate:
• S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
• S₁ > 0 → outside the circle
• S₁ = 0 → on the circle
• S₁ < 0 → inside the circle

Position of a Line w.r.t. a Circle

• If p = perpendicular distance from centre to line, r = radius:
• p > r → line outside circle
• p = r → tangent
• p < r → secant (chord)
• p = 0 → diameter

Equation of Tangent

• Tangent at point (x₁, y₁) to x² + y² + 2gx + 2fy + c = 0:
• xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
• Tangent to x² + y² = a² with slope m:
• y = mx ± a√(1 + m²)

Equation of Normal

• Normal at (x₁, y₁) to x² + y² + 2gx + 2fy + c = 0:
• (x − x₁)/(x₁ + g) = (y − y₁)/(y₁ + f)
• For x² + y² = a²:
• x/x₁ = y/y₁

Length of Tangent from a Point

• From point (x₁, y₁) to circle S = 0:
• Length = √S₁
• Angle between two tangents:
• 2 tan⁻¹ (√S₁ / r)

Pair of Tangents

• Combined equation from point (x₁, y₁):
• S · S₁ = T²

Director Circle

• Locus of intersection of perpendicular tangents
• For x² + y² = a²:
• x² + y² = 2a²

Chord of Contact

• From point (x₁, y₁) to circle S = 0:
• xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

Pole & Polar

• Polar of point (x₁, y₁) w.r.t. x² + y² = a²:
• xx₁ + yy₁ = a²
• For general circle:
• xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

Radical Axis

• For circles S = 0 and S′ = 0:
• S − S′ = 0
• Perpendicular to line joining centres
• Coincides with common chord if circles intersect

Radical Centre

• Point of intersection of radical axes of three circles
• Obtained by solving any two of:
• S₁ − S₂ = 0, S₂ − S₃ = 0, S₃ − S₁ = 0

Family of Circles

• Through intersection of two circles:
• S + λS′ = 0, λ ≠ −1
• Through intersection of a circle and a line:
• S + λL = 0