Hyperbola

Transverse axis along the x-axis, length 2a. Conjugate axis along the y-axis, length 2b. Vertices at (±a, 0). Foci at (±c, 0) where c² = a² + b².

Transverse axis along the y-axis, length 2a. Vertices at (0, ±a). Foci at (0, ±c) where c² = a² + b². Asymptotes: y = ±(a/b)x.

Centre at (h, k). Identified by completing the square. All elements of the standard hyperbola apply with origin shifted to (h, k).

Unlike the ellipse where c² = a² − b², for a hyperbola c² = a² + b² (the foci are farther from the centre than the vertices). b² is not less than a²; a and b have no size constraint relative to each other.

Eccentricity of a hyperbola is always greater than 1. e = √2 for a rectangular hyperbola (a = b). e → 1⁺ gives a very elongated hyperbola; e → ∞ gives one approaching a pair of lines. Directrices at x = ±a/e (between vertices and centre).

For a point P(x₁, y₁) on the right branch (x₁ > 0) of x²/a² − y²/b² = 1: distance to nearer focus S′(c,0) is ex₁ − a, and to farther focus S(−c,0) is ex₁ + a. On the left branch (x₁ < 0), the nearer focus is S(−c,0) and the distances reverse.

The defining property of a hyperbola: the absolute difference of distances from any point on the hyperbola to the two foci is constant, equal to the transverse axis length 2a. Points on the right branch satisfy r₂ − r₁ = 2a; on the left branch r₁ − r₂ = 2a.

Each focus has a latus rectum of length 2b²/a, the same formula as the ellipse and parabola. Endpoints of the latus rectum through (c, 0) are (c, ±b²/a).

The two asymptotes of x²/a² − y²/b² = 1. The hyperbola approaches but never reaches them. Combined equation: x²/a² − y²/b² = 0. The asymptotes pass through the centre and make equal angles with both axes.

The conjugate of x²/a² − y²/b² = 1 is x²/a² − y²/b² = −1, equivalently y²/b² − x²/a² = 1. They share the same asymptotes. If e₁ and e₂ are eccentricities of a hyperbola and its conjugate: 1/e₁² + 1/e₂² = 1.

Every point on x²/a² − y²/b² = 1 can be written as (a secθ, b tanθ). θ ∈ [0°, 360°) with θ ≠ 90°, 270°. The right branch corresponds to θ ∈ (−90°, 90°) and the left branch to θ ∈ (90°, 270°).

Tangent to x²/a² − y²/b² = 1 at point (x₁, y₁) lying on it. Obtained by the T = 0 rule: replace x² → xx₁, y² → yy₁.

Tangent at (a secθ, b tanθ). Slope = (b secθ)/(a tanθ) = b/(a sinθ).

Tangents to x²/a² − y²/b² = 1 with slope m, valid only when |m| > b/a (otherwise the line is parallel to or between the asymptotes and no tangent exists). Point of contact: (±a²m/√(a²m²−b²), ±b²/√(a²m²−b²)).

Line y = mx + c is tangent to x²/a² − y²/b² = 1 iff c² = a²m² − b². Requires |m| > b/a. Note the sign difference from the ellipse (where c² = a²m² + b²).

The tangent at any point P on the hyperbola meets the two asymptotes at Q and R. P is the midpoint of QR, and the area of triangle OQR (O = centre) is constant = ab, independent of the choice of P.

Normal to x²/a² − y²/b² = 1 at (x₁, y₁). Note the + signs on both terms (contrast with the ellipse normal which has a − sign).

Normal at (a secθ, b tanθ). Slope of normal = −a sinθ/b.

Chord joining the two tangent contact points from external point (x₁, y₁). Same form as the tangent at a point.

Combined equation of tangents from (x₁, y₁) where S = x²/a² − y²/b² − 1, S₁ = x₁²/a² − y₁²/b² − 1, T = xx₁/a² − yy₁/b² − 1.

Chord of x²/a² − y²/b² = 1 whose midpoint is (x₁, y₁). Explicitly: xx₁/a² − yy₁/b² − 1 = x₁²/a² − y₁²/b² − 1.

Locus of points from which the two tangents to x²/a² − y²/b² = 1 are perpendicular. Real only when a > b. When a = b (rectangular hyperbola), the director circle degenerates to a point (the centre). When a < b, no real director circle exists.

Polar of (x₁, y₁) with respect to x²/a² − y²/b² = 1. Same form as the chord of contact and the tangent at a point. La Hire's theorem holds.

Special case when the asymptotes are perpendicular (a = b, eccentricity e = √2). The form xy = c² arises from x²/c² − y²/c² = 1 rotated 45°. Asymptotes are the coordinate axes.

Every point on xy = c² is (ct, c/t) for a unique t ∈ ℝ\{0}. t > 0 gives the first quadrant branch; t < 0 gives the third quadrant branch.

Tangent to xy = c² at (ct, c/t). Slope = −1/t². Intercepts: x-intercept 2ct, y-intercept 2c/t. The tangent at t meets the asymptotes (axes) at (2ct, 0) and (0, 2c/t), and the point (ct, c/t) is the midpoint of this segment.

Normal to xy = c² at (ct, c/t). Slope = t². Up to four normals can be drawn from a general point to xy = c².