FORMULA LIBRARY : HYPERBOLA

HYPERBOLA

— JEE MATH APEX —

JEE MATH APEX

1. Standard Equation of Hyperbola

• Horizontal hyperbola: x²/a² − y²/b² = 1
• Centre: (0, 0)
• Transverse axis: x-axis (length 2a)
• Conjugate axis: y-axis (length 2b)
• Vertical hyperbola: y²/a² − x²/b² = 1
• Vertices: (±a, 0), Foci: (±ae, 0)

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2. Key Parameters

• Eccentricity: e = √(1 + b²/a²)  (e > 1)
• b² = a²(e² − 1)
• Foci: (±ae, 0)
• Directrices: x = ±a/e
• Latus rectum: 2b²/a
• Ends of latus rectum: (±ae, ±b²/a)

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3. Parametric Form

• For x²/a² − y²/b² = 1: x = a sec θ, y = b tan θ
• Parametric point: (a sec θ, b tan θ)
• θ is called eccentric angle
• Auxiliary circle: x² + y² = a²
• Alternative: x = a cosh t, y = b sinh t

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4. Tangent to Hyperbola

• At point (x₁,y₁): xx₁/a² − yy₁/b² = 1
• At parametric point (a sec θ, b tan θ): (x sec θ)/a − (y tan θ)/b = 1
• With slope m: y = mx ± √(a²m² − b²)
• Point of contact: (±a²m/√(a²m²−b²), ∓b²/√(a²m²−b²))
• Condition for tangency: c = ±√(a²m² − b²)

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5. Normal to Hyperbola

• At point (x₁,y₁): a²x/x₁ + b²y/y₁ = a² + b²
• At parametric point (a sec θ, b tan θ): ax cos θ + by cot θ = a² + b²
• With slope m: y = mx ± m(a²+b²)/√(a²−b²m²)
• Four normals can be drawn from a point to hyperbola

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6. Asymptotes

• Asymptotes of x²/a² − y²/b² = 1: y = ±(b/a)x
• Joint equation: x²/a² − y²/b² = 0
• Angle between asymptotes: 2 tan⁻¹(b/a)
• If asymptotes are perpendicular (rectangular hyperbola): a = b

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7. Rectangular Hyperbola

• Equation: x² − y² = a²
• Eccentricity: e = √2
• Asymptotes are perpendicular: y = ±x
• Referred to asymptotes: xy = c² (where c² = a²/2)
• Parametric form: x = ct, y = c/t

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8. Chord of Contact

• Chord of contact from (x₁,y₁): xx₁/a² − yy₁/b² = 1
• Pair of tangents: SS₁ = T²
• Where S = x²/a² − y²/b² − 1, S₁ = x₁²/a² − y₁²/b² − 1, T = xx₁/a² − yy₁/b² − 1

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9. Chord with Midpoint

• Chord with midpoint (x₁,y₁): T = S₁
• xx₁/a² − yy₁/b² − 1 = x₁²/a² − y₁²/b² − 1
• Slope of chord with midpoint (x₁,y₁): b²x₁/(a²y₁)

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10. Director Circle

• Director circle of x²/a² − y²/b² = 1: x² + y² = a² − b²
• Exists only if a > b
• Locus of intersection of perpendicular tangents
• If a = b (rectangular hyperbola): Director circle reduces to point

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11. Focal Distances

• For any point P on hyperbola: |SP − S’P| = 2a (constant)
• Focal distance of point (x₁,y₁): SP = ex₁ − a and S’P = ex₁ + a
• Difference of focal distances = Transverse axis length = 2a

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12. Position of a Point

• For hyperbola x²/a² − y²/b² = 1 and point (x₁,y₁):
• If x₁²/a² − y₁²/b² − 1 < 0: Point outside
• If x₁²/a² − y₁²/b² − 1 = 0: Point on hyperbola
• If x₁²/a² − y₁²/b² − 1 > 0: Point inside

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13. Conjugate Hyperbola

• Conjugate of x²/a² − y²/b² = 1 is: x²/a² − y²/b² = −1 or y²/b² − x²/a² = 1
• Hyperbola and its conjugate have same asymptotes
• Eccentricity of conjugate: e’ = √(1 + a²/b²)
• Relation: 1/e² + 1/e’² = 1

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14. Important Theorems (Advanced)

• If normal at θ₁ meets hyperbola again at θ₂: tan(θ₁/2)·tan(θ₂/2) = −(a+b)/(a−b)
• Tangents at extremities of focal chord intersect on directrix
• Product of perpendiculars from foci to any tangent = b²
• Area of triangle formed by asymptotes and tangent = ab (constant)
• For rectangular hyperbola xy = c², equation of chord: x(t₁+t₂) + y(t₁t₂) = c(t₁+t₂)

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15. Tangent & Normal for xy = c²

• Tangent at (ct, c/t): x/t + yt = 2c
• Normal at (ct, c/t): xt³ − yt − ct⁴ + c = 0
• Chord joining t₁ and t₂: x + yt₁t₂ = c(t₁ + t₂)
• Foci: (±c√2, ±c√2)
• Eccentricity: e = √2