Cheat Sheet Hyperbola PDF

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Summary

Hyberbola formulae....

Description

DOUBLEROOT

Cheat Sheet – Hyperbola Chord

Equations Focus: (±ae, 0), Directrix: x = ± a/e x2 a2

−

y2 b2

=1

(Standard)

where b2 = a2 (e2 − 1)

where b2 = a2 (e2 − 1)

Focus: (x1, y1), Directrix: ax + by + c = 0 (ax + by + c)2 (x − x1 )2 + (y − y1 )2 = e2 a2 + b2 where e → eccentricity, e > 1

Parametric Equation x = a sec ϕ y = b tan ϕ

ϕ1 + ϕ2 ϕ1 − ϕ2 y ϕ1 + ϕ2 x − sin = cos cos 2 2 a b 2

Chord of contact w.r.t the point (x1, y1)

Focus: (0, ±ae), Directrix: y = ± a/e y2 x2 − =1 a2 b 2

Chord with end points (φ1) and (φ2)

where ϕ → eccentric angle

T=0

Chord with mid-point (x1, y1)

x2 y2 −1 − a2 b 2 xx1 yy1 T= 2 − 2 −1 b a x12 y12 S1 = 2 − 2 − 1 b a

S=

(S1 > 0)

T = S1

Asymptotes (Standard) Equation: x2

y2 =0 − a2 b 2

Angle between the asymptotes: θ = 2 tan−1

Notations (Standard)

(S1 < 0)

b a

Some Properties of the Hyperbola

Position of a point (x1, y1) w.r.t. the hyperbola Outside: S1 < 0, On: S1 = 0, Inside: S1 > 0

Tangent Equation of the tangent having slope m y = mx ± √a2 m2 − b2

Equation of the tangent at the point (x1, y1) T=0

(S1 = 0)

Equation of the tangent at the point (φ) y x sec ϕ − tan ϕ − 1 = 0 a b

Pair of tangents from an external point (x1, y1) SS1 = T2

(S1 < 0)

Normal

>> Difference of the focal distances of any point on the hyperbola is constant (|PS - PS’| = 2a) >> Harmonic mean of the segments of any focal chord is equal to the semi latus rectum (1/PS + 1/QS = 2a/b2)

Equation of the normal at the point (x1, y1) x − x1 y − y1 x = y1 a2 −b2

>> Segment of tangent intercepted between point of contact and the directrix subtends right angle at focus (∠KSP=90°)

Equation of the normal at the point (φ) ax cos ϕ + by cot ϕ = a2 + b2

>> Feet of perpendicular from the foci upon any tangent lie on the auxiliary circle (SN⊥PN, S’N’⊥P’N’)

Equation of the normal having slope m y = mx ±

(a2 + b2 )m √a2 − b2 m2

>> Product of the lengths of perpendiculars from the foci upon any tangent is constant (SN x S’N’ = b2) >> Tangent at any point P bisects ∠SPS’

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