Hyperbola

Standard Equation of a Hyperbola

• x²/a² − y²/b² = 1 (transverse axis along x-axis)
• y²/a² − x²/b² = 1 (transverse axis along y-axis)
• Centre: (0, 0)

Symmetry

• Symmetric about x-axis
• Symmetric about y-axis
• Symmetric about origin

Axes & Key Lengths

• Length of transverse axis: 2a
• Length of conjugate axis: 2b
• Vertices: (±a, 0)
• Conjugate vertices: (0, ±b)

Foci & Eccentricity

• Distance of focus from centre: ae
• Foci: (±ae, 0)
• Distance between foci: 2ae
• b² = a²(e² − 1)
• e = √(1 + b²/a²)

Directrices

• Equations: x = ±a/e
• Distance between directrices: 2a/e

Important Properties

• For any point P on hyperbola: S′P − SP = 2a
• All chords passing through centre are bisected at centre

Latus Rectum

• Length: 2b²/a
• Equations: x = ±ae
• Endpoints: (±ae, ±b²/a)

Parametric Coordinates

• x = a secθ
• y = b tanθ
• Point corresponding to parameter θ is called point θ

Equation of Chord

• Chord joining points θ₁ and θ₂:
• (x/a) sec((θ₁−θ₂)/2) − (y/b) tan((θ₁+θ₂)/2) = 1

Position of a Point

• For point (x₁, y₁), evaluate:
• x₁²/a² − y₁²/b²
• > 1 → outside hyperbola
• = 1 → on hyperbola
• < 1 → inside hyperbola

Equation of Tangent

Point Form

• Tangent at (x₁, y₁): xx₁/a² − yy₁/b² = 1

Parametric Form

• Tangent at (a secθ, b tanθ): (x/a) secθ − (y/b) tanθ = 1

Slope Form

• y = mx ± √(a²m² − b²)
• Condition of tangency: c² = a²m² − b²

Director Circle

• Locus of intersection of perpendicular tangents
• Equation: x² + y² = a² − b²

Equation of Normal

Point Form

• a²x/x₁ + b²y/y₁ = a² + b²

Parametric Form

• ax/secθ + by/tanθ = a² + b²

Slope Form

• y = mx ± m(a² + b²)/√(a² − b²m²)

Pair of Tangents from a Point

• SS₁ = T²
• Where S = x²/a² − y²/b² − 1, S₁ = x₁²/a² − y₁²/b² − 1, T = xx₁/a² − yy₁/b² − 1

Chord with Given Midpoint

• Equation: T = S₁

Chord of Contact

• Equation: T = 0

Pole & Polar

• Polar of point (x₁, y₁): T = 0
• Polar of focus is the directrix

Diameter of a Hyperbola

• Diameter bisecting chords of slope m:
• y = (b²/a²m) x

Conjugate Diameters

• If slopes are m₁, m₂:
• m₁m₂ = b²/a²

Asymptotes

• Equations: y = ±(b/a)x
• Angle between asymptotes: 2 tan⁻¹(b/a)
• Asymptotes pass through centre
• Hyperbola and its conjugate have same asymptotes

Rectangular Hyperbola

• If asymptotes are perpendicular ⇒ a = b
• Equation: x² − y² = a²
• With axes along asymptotes: xy = c²

Parametric Form (Rectangular Hyperbola)

• x = ct, y = c/t

Important Results (Rectangular Hyperbola)

• Asymptotes: y = ±x
• Slope of tangent at (ct, c/t): −1/t²
• Slope of normal at (ct, c/t): t²
• Equation of tangent at point (x₁, y₁): xy₁ + x₁y = 2c²