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Hyperbola formulas

Master Hyperbola through 36 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Hyperbola, every formula

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Standard equation
$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$
Q1MCQStandard form
$\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$ has $a^{2}=$:
- A
  $16$
- B
  $9$
- C
  $25$
- D
  $7$
Eccentricity
$e = \frac{a ^{2} + b ^{2}}{a}, b^{2} = a^{2} (e^{2} - 1)$
Q1MCQEccentricity
For $\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$ ($a=4,b=3$), the eccentricity is:
- A
  $\dfrac54$
- B
  $\dfrac45$
- C
  $\dfrac53$
- D
  $\dfrac34$
Eccentricity condition
$e > 1$
Q1MCQe condition
For a hyperbola, the eccentricity satisfies:
- A
  $e>1$
- B
  $e=1$
- C
  $e<1$
- D
  $e=0$
Foci
$(\pm a e, 0)$
Q1MCQFoci
The foci of $\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$ (e=5/4) are:
- A
  $(\pm5,0)$
- B
  $(\pm4,0)$
- C
  $(0,\pm5)$
- D
  $(\pm3,0)$
Vertices
$(\pm a, 0)$
Q1MCQVertices
The vertices of $\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$ are:
- A
  $(\pm4,0)$
- B
  $(\pm3,0)$
- C
  $(0,\pm4)$
- D
  $(\pm5,0)$
Length of latus rectum
$\frac{2 b ^{2}}{a}$
Transverse & conjugate axes
$2 a (transverse), 2 b (conjugate)$
Directrices
$x = \pm \frac{a}{e}$
Q1MCQDirectrices
The directrices of $\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$ (e=5/4) are:
- A
  $x=\pm\dfrac{16}{5}$
- B
  $x=\pm5$
- C
  $x=\pm4$
- D
  $y=\pm\dfrac{16}{5}$
Ends of latus rectum
$(\pm a e, \pm \frac{b ^{2}}{a})$
Q1MCQEnds of LR
The ends of a latus rectum of $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ are:
- A
  $\left(\pm ae,\pm\dfrac{b^{2}}{a}\right)$
- B
  $(\pm a,\pm b)$
- C
  $(\pm ae,\pm b)$
- D
  $\left(\pm a,\pm\dfrac{b^{2}}{a}\right)$
Focal distance difference
$∣ S P - S^{'} P ∣ = 2 a$
Q1MCQFocal difference
For any point $P$ on a hyperbola with foci $S,S'$:
- A
  $|SP-S'P|=2a$
- B
  $SP+S'P=2a$
- C
  $|SP-S'P|=2b$
- D
  $SP\cdot S'P=a^{2}$
Conjugate hyperbola
$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = - 1$
Q1MCQConjugate
The conjugate of $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=-1$
- B
  $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$
- C
  $\dfrac{y^{2}}{a^{2}}-\dfrac{x^{2}}{b^{2}}=1$
- D
  $\dfrac{x^{2}}{b^{2}}-\dfrac{y^{2}}{a^{2}}=1$
Eccentricities relation
$\frac{1}{e _{1}^{2}} + \frac{1}{e _{2}^{2}} = 1$
Q1MCQEccentricities
For a hyperbola and its conjugate, $\dfrac{1}{e_1^{2}}+\dfrac{1}{e_2^{2}}$ equals:
- A
  $1$
- B
  $2$
- C
  $0$
- D
  $e_1e_2$
Parametric equations
$x = a sec θ, y = b tan θ$
Q1MCQParametric
A parametric point on $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $(a\sec\theta,b\tan\theta)$
- B
  $(a\cos\theta,b\sin\theta)$
- C
  $(a\tan\theta,b\sec\theta)$
- D
  $(a\theta,b\theta)$
Auxiliary circle
$x^{2} + y^{2} = a^{2}$
Q1MCQAuxiliary circle
The auxiliary circle of $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $x^{2}+y^{2}=a^{2}$
- B
  $x^{2}+y^{2}=b^{2}$
- C
  $x^{2}+y^{2}=a^{2}+b^{2}$
- D
  $x^{2}+y^{2}=a^{2}-b^{2}$
Director circle
$x^{2} + y^{2} = a^{2} - b^{2}$
Q1MCQDirector circle
The director circle of $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $x^{2}+y^{2}=a^{2}-b^{2}$
- B
  $x^{2}+y^{2}=a^{2}+b^{2}$
- C
  $x^{2}+y^{2}=a^{2}$
- D
  $x^{2}+y^{2}=b^{2}$
Rectangular hyperbola
$a = b \Rightarrow e = 2$
Q1MCQRectangular
The eccentricity of a rectangular hyperbola ($a=b$) is:
- A
  $\sqrt2$
- B
  $2$
- C
  $1$
- D
  $\dfrac{1}{\sqrt2}$
Notation S₁₁
$S_{11} = \frac{x _{1}^{2}}{a ^{2}} - \frac{y _{1}^{2}}{b ^{2}} - 1$
Q1MCQS₁₁
For $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$, $S_{11}$ equals:
- A
  $\dfrac{x_1^{2}}{a^{2}}-\dfrac{y_1^{2}}{b^{2}}-1$
- B
  $\dfrac{x_1^{2}}{a^{2}}+\dfrac{y_1^{2}}{b^{2}}-1$
- C
  $\dfrac{x_1^{2}}{a^{2}}-\dfrac{y_1^{2}}{b^{2}}$
- D
  $\dfrac{xx_1}{a^{2}}-\dfrac{yy_1}{b^{2}}-1$
Position of a point
$S_{11} < 0 outside, = 0 on, > 0 inside$
Q1MCQPosition
If $S_{11}>0$ for a point, it lies:
- A
  inside the hyperbola
- B
  outside
- C
  on the hyperbola
- D
  at the centre
Tangent at (x₁,y₁)
$\frac{x x _{1}}{a ^{2}} - \frac{y y _{1}}{b ^{2}} = 1$
Q1MCQTangent at point
The tangent to $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ at $(x_1,y_1)$ is:
- A
  $\dfrac{xx_1}{a^{2}}-\dfrac{yy_1}{b^{2}}=1$
- B
  $\dfrac{xx_1}{a^{2}}+\dfrac{yy_1}{b^{2}}=1$
- C
  $\dfrac{xx_1}{a^{2}}-\dfrac{yy_1}{b^{2}}=0$
- D
  $y=mx\pm\sqrt{a^{2}m^{2}-b^{2}}$
Tangent at θ
$\frac{x}{a} sec θ - \frac{y}{b} tan θ = 1$
Q1MCQTangent at θ
The tangent at parameter $\theta$ is:
- A
  $\dfrac{x}{a}\sec\theta-\dfrac{y}{b}\tan\theta=1$
- B
  $\dfrac{x}{a}\cos\theta+\dfrac{y}{b}\sin\theta=1$
- C
  $\dfrac{x}{a}\tan\theta-\dfrac{y}{b}\sec\theta=1$
- D
  $\dfrac{xx_1}{a^{2}}-\dfrac{yy_1}{b^{2}}=1$
Tangent of slope m
$y = m x \pm a^{2} m^{2} - b^{2}$
Q1MCQTangent slope m
The tangent of slope $m$ to $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $y=mx\pm\sqrt{a^{2}m^{2}-b^{2}}$
- B
  $y=mx\pm\sqrt{a^{2}m^{2}+b^{2}}$
- C
  $y=mx\pm\dfrac{a}{m}$
- D
  $y=mx\pm b$
Tangency condition
$c^{2} = a^{2} m^{2} - b^{2}$
for y=mx+c
Q1MCQTangency condition
$y=mx+c$ touches $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ iff:
- A
  $c^{2}=a^{2}m^{2}-b^{2}$
- B
  $c^{2}=a^{2}m^{2}+b^{2}$
- C
  $c=\dfrac{a}{m}$
- D
  $c^{2}=a^{2}+b^{2}m^{2}$
Normal at (x₁,y₁)
$\frac{a ^{2} x}{x _{1}} + \frac{b ^{2} y}{y _{1}} = a^{2} + b^{2}$
Q1MCQNormal at point
The normal to $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ at $(x_1,y_1)$ is:
- A
  $\dfrac{a^{2}x}{x_1}+\dfrac{b^{2}y}{y_1}=a^{2}+b^{2}$
- B
  $\dfrac{a^{2}x}{x_1}-\dfrac{b^{2}y}{y_1}=a^{2}+b^{2}$
- C
  $\dfrac{xx_1}{a^{2}}-\dfrac{yy_1}{b^{2}}=1$
- D
  $a^{2}x+b^{2}y=1$
Normal of slope m
$y = m x \mp \frac{m ( a ^{2} + b ^{2} )}{a ^{2} - b ^{2} m ^{2}}$
Q1MCQNormal slope m
The slope-$m$ normal to $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $y=mx\mp\dfrac{m(a^{2}+b^{2})}{\sqrt{a^{2}-b^{2}m^{2}}}$
- B
  $y=mx\pm\sqrt{a^{2}m^{2}-b^{2}}$
- C
  $y=mx\pm\dfrac{a}{m}$
- D
  $y=mx\mp m(a^{2}+b^{2})$
Asymptotes
$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 0 \Rightarrow y = \pm \frac{b}{a} x$
Q1MCQAsymptotes
The asymptotes of $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ are:
- A
  $y=\pm\dfrac{b}{a}x$
- B
  $y=\pm\dfrac{a}{b}x$
- C
  $y=\pm x$
- D
  $x=\pm a$
Angle between asymptotes
$2 tan^{- 1} \frac{b}{a}$
Q1MCQAngle between asymptotes
The angle between the asymptotes is:
- A
  $2\tan^{-1}\dfrac{b}{a}$
- B
  $\tan^{-1}\dfrac{b}{a}$
- C
  $2\tan^{-1}\dfrac{a}{b}$
- D
  $90^{\circ}$ always
Hyperbola - asymptote constant
$hyperbola and its asymptote pair differ by a constant$
Q1MCQHyperbola vs asymptote
A hyperbola and its pair of asymptotes differ by:
- A
  a constant
- B
  a linear term
- C
  a quadratic term
- D
  nothing
S + S' = 2A
$hyperbola + conjugate = 2 \times asymptote pair$
Q1MCQS+S'=2A
If $S$, $S'$, $A$ are the hyperbola, its conjugate and its asymptote pair, then:
- A
  $S+S'=2A$
- B
  $S+S'=A$
- C
  $S-S'=2A$
- D
  $SS'=A$
Rectangular hyperbola xy=c²
$x y = c^{2}, c^{2} = \frac{a ^{2}}{2}$
Q1MCQxy=c²
For the rectangular hyperbola $xy=c^{2}$, $c^{2}$ relates to $a$ as:
- A
  $c^{2}=\dfrac{a^{2}}{2}$
- B
  $c^{2}=a^{2}$
- C
  $c^{2}=2a^{2}$
- D
  $c^{2}=\dfrac{a}{2}$
xy=c² parametric
$x = c t, y = \frac{c}{t}$
Q1MCQxy=c² parametric
A parametric point on $xy=c^{2}$ is:
- A
  $\left(ct,\dfrac{c}{t}\right)$
- B
  $(c\sec t,c\tan t)$
- C
  $(ct^{2},2ct)$
- D
  $(c\cos t,c\sin t)$
xy=c² tangent at t
$x + y t^{2} = 2 c t$
Q1MCQxy=c² tangent
The tangent to $xy=c^{2}$ at parameter $t$ is:
- A
  $x+yt^{2}=2ct$
- B
  $x-yt^{2}=2ct$
- C
  $xt+y=2c$
- D
  $xt^{3}-yt=c(t^{4}-1)$
xy=c² normal at t
$x t^{3} - y t = c (t^{4} - 1)$
Q1MCQxy=c² normal
The normal to $xy=c^{2}$ at parameter $t$ is:
- A
  $xt^{3}-yt=c(t^{4}-1)$
- B
  $x+yt^{2}=2ct$
- C
  $xt-y=c$
- D
  $x+yt=2ct$
Chord of contact
$\frac{x x _{1}}{a ^{2}} - \frac{y y _{1}}{b ^{2}} = 1 (S_{1} = 0)$
Q1MCQChord of contact
The chord of contact of $(x_1,y_1)$ w.r.t. $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $\dfrac{xx_1}{a^{2}}-\dfrac{yy_1}{b^{2}}=1$
- B
  $\dfrac{xx_1}{a^{2}}+\dfrac{yy_1}{b^{2}}=1$
- C
  $S_{11}=0$
- D
  $y=mx\pm\sqrt{a^{2}m^{2}-b^{2}}$
Product of ⊥ from foci to tangent
$b^{2}$
Q1MCQProduct of ⊥ from foci
The product of perpendiculars from the two foci to any tangent is:
- A
  $b^{2}$
- B
  $a^{2}$
- C
  $a^{2}+b^{2}$
- D
  $ab$
Reflection property
$ray toward one focus reflects toward the other focus$
Q1MCQReflection
A ray directed toward one focus of a hyperbola reflects toward:
- A
  the other focus
- B
  the centre
- C
  the directrix
- D
  infinity
Co-normal points sum
$α + β + γ + δ = odd multiple of π$
eccentric angles
Q1MCQCo-normal points
The sum of the eccentric angles of the four co-normal points on a hyperbola is:
- A
  an odd multiple of $\pi$
- B
  an even multiple of $\pi$
- C
  $0$
- D
  $\pi/2$

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Hyperbola formulas

Hyperbola, every formula

Standard equation

Eccentricity

Eccentricity condition

Foci

Vertices

Length of latus rectum

Transverse & conjugate axes

Directrices

Ends of latus rectum

Focal distance difference

Conjugate hyperbola

Eccentricities relation

Parametric equations

Auxiliary circle

Director circle

Rectangular hyperbola

Notation S₁₁

Position of a point

Tangent at (x₁,y₁)

Tangent at θ

Tangent of slope m

Tangency condition

Normal at (x₁,y₁)

Normal of slope m

Asymptotes

Angle between asymptotes

Hyperbola - asymptote constant

S + S' = 2A

Rectangular hyperbola xy=c²

xy=c² parametric

xy=c² tangent at t

xy=c² normal at t

Chord of contact

Product of ⊥ from foci to tangent

Reflection property

Co-normal points sum

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Other ways to revise this chapter

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