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Ellipse formulas

Master Ellipse through 34 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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Ellipse, every formula

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Standard equation
$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$
Q1MCQStandard form
$\dfrac{x^{2}}{25}+\dfrac{y^{2}}{9}=1$ has $a^{2}=$:
- A
  $25$
- B
  $9$
- C
  $16$
- D
  $34$
Eccentricity (a>b)
$e = 1 - \frac{b ^{2}}{a ^{2}}, b^{2} = a^{2} (1 - e^{2})$
Q1MCQEccentricity
For $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{9}=1$ ($a=5,b=3$), the eccentricity is:
- A
  $\dfrac45$
- B
  $\dfrac35$
- C
  $\dfrac54$
- D
  $\dfrac25$
Eccentricity condition
$0 < e < 1$
Q1MCQe condition
For an ellipse, the eccentricity satisfies:
- A
  $0<e<1$
- B
  $e=1$
- C
  $e>1$
- D
  $e=0$
Foci
$(\pm a e, 0)$
Q1MCQFoci
The foci of $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{9}=1$ (e=4/5) are:
- A
  $(\pm4,0)$
- B
  $(\pm5,0)$
- C
  $(0,\pm4)$
- D
  $(\pm3,0)$
Vertices
$(\pm a, 0)$
Q1MCQVertices
The vertices of $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{9}=1$ are:
- A
  $(\pm5,0)$
- B
  $(\pm3,0)$
- C
  $(0,\pm5)$
- D
  $(\pm4,0)$
Directrices
$x = \pm \frac{a}{e}$
Q1MCQDirectrices
The directrices of $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{9}=1$ (e=4/5) are:
- A
  $x=\pm\dfrac{25}{4}$
- B
  $x=\pm4$
- C
  $x=\pm5$
- D
  $y=\pm\dfrac{25}{4}$
Length of major & minor axes
$2 a (major), 2 b (minor)$
Length of latus rectum
$\frac{2 b ^{2}}{a}$
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
  $\left(\pm a,\pm\dfrac{b^{2}}{a}\right)$
Sum of focal distances
$S P + S^{'} P = 2 a$
Q1MCQSum of focal distances
For any point $P$ on an ellipse, $SP+S'P$ equals:
- A
  $2a$
- B
  $2b$
- C
  $2ae$
- D
  $a+b$
Focal distances
$S P = a - e x_{1}, S^{'} P = a + e x_{1}$
Q1MCQFocal distance
For $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, $SP$ equals:
- A
  $a-ex_1$
- B
  $a+ex_1$
- C
  $ex_1-a$
- D
  $2a$
Distance between foci
$S S^{'} = 2 a e$
Parametric point
$(a cos θ, b sin θ)$
Q1MCQParametric
A parametric point on $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $(a\cos\theta,b\sin\theta)$
- B
  $(a\sec\theta,b\tan\theta)$
- C
  $(at^{2},2at)$
- D
  $(a\sin\theta,b\cos\theta)$
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
$S_{11} < 0 inside, = 0 on, > 0 outside$
Q1MCQPosition
If $S_{11}<0$ for a point, it lies:
- A
  inside the ellipse
- B
  outside
- C
  on the ellipse
- D
  at a focus
Focal chord (semi-LR = HM)
$\frac{1}{S P} + \frac{1}{S ^{'} P} = \frac{2}{S L}$
Q1MCQHM of focal segments
For a focal chord, $\dfrac{1}{SP}+\dfrac{1}{S'P}$ equals:
- A
  $\dfrac{2}{SL}$ (semi-LR is their HM)
- B
  $\dfrac{1}{a}$
- C
  $\dfrac{2}{a}$
- D
  $a$
Chord joining α,β
$\frac{x}{a} cos \frac{α + β}{2} + \frac{y}{b} sin \frac{α + β}{2} = cos \frac{α - β}{2}$
Q1MCQChord α,β
The chord joining $\alpha,\beta$ on the ellipse is:
- A
  $\dfrac{x}{a}\cos\dfrac{\alpha+\beta}{2}+\dfrac{y}{b}\sin\dfrac{\alpha+\beta}{2}=\cos\dfrac{\alpha-\beta}{2}$
- B
  $\dfrac{x}{a}\cos\dfrac{\alpha-\beta}{2}=1$
- C
  $\dfrac{xx_1}{a^{2}}+\dfrac{yy_1}{b^{2}}=1$
- D
  $y=mx\pm\sqrt{a^{2}m^{2}+b^{2}}$
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
$\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1$
Q1MCQTangent at θ
The tangent at parameter $\theta$ is:
- A
  $\dfrac{x}{a}\cos\theta+\dfrac{y}{b}\sin\theta=1$
- B
  $\dfrac{x}{a}\sec\theta-\dfrac{y}{b}\tan\theta=1$
- C
  $\dfrac{x}{a}\sin\theta+\dfrac{y}{b}\cos\theta=1$
- 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
  $c^{2}=a^{2}+b^{2}m^{2}$
Point of contact (slope form)
$(- \frac{a ^{2} m}{c}, \frac{b ^{2}}{c})$
Q1MCQPoint of contact
The point of contact of the slope-$m$ tangent ($y=mx+c$) to the ellipse is:
- A
  $\left(-\dfrac{a^{2}m}{c},\dfrac{b^{2}}{c}\right)$
- B
  $\left(\dfrac{a^{2}m}{c},\dfrac{b^{2}}{c}\right)$
- C
  $\left(\dfrac{a^{2}}{m^{2}},\dfrac{2a}{m}\right)$
- D
  $(am^{2},-2am)$
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 at θ
$\frac{a x}{cos θ} - \frac{b y}{sin θ} = a^{2} - b^{2}$
Q1MCQNormal at θ
The normal at parameter $\theta$ is:
- A
  $\dfrac{ax}{\cos\theta}-\dfrac{by}{\sin\theta}=a^{2}-b^{2}$
- B
  $\dfrac{ax}{\cos\theta}+\dfrac{by}{\sin\theta}=a^{2}+b^{2}$
- C
  $\dfrac{x}{a}\cos\theta+\dfrac{y}{b}\sin\theta=1$
- D
  $ax\cos\theta-by\sin\theta=a^{2}-b^{2}$
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. the ellipse 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
  $y=mx\pm\sqrt{a^{2}m^{2}+b^{2}}$
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
  $x^{2}+y^{2}=b^{2}$
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
  $x^{2}+y^{2}=a^{2}-b^{2}$
Area of ellipse
$π ab$
Q1MCQArea
The area of $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{9}=1$ is:
- A
  $15\pi$
- B
  $225\pi$
- C
  $34\pi$
- D
  $8\pi$
Max inscribed rectangle
$2 ab, sides a 2, b 2$
Q1MCQMax rectangle
The maximum rectangle inscribed in $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ has area:
- A
  $2ab$
- B
  $ab$
- C
  $4ab$
- D
  $\pi ab$
Product of ⊥ from foci to tangent
$b^{2}$
Q1MCQProduct of ⊥ from foci
The product of perpendiculars from the foci to any tangent (a>b) is:
- A
  $b^{2}$
- B
  $a^{2}$
- C
  $a^{2}+b^{2}$
- D
  $ab$
Midpoint chord
$S_{1} = S_{11}$
Q1MCQMidpoint chord
The chord of the ellipse with midpoint $(x_1,y_1)$ is:
- A
  $S_1=S_{11}$
- B
  $S_1=0$
- C
  $S_{11}=0$
- D
  $S=0$
Diameter (bisecting slope-m chords)
$y = - \frac{b ^{2}}{a ^{2} m} x$
Q1MCQDiameter
The diameter bisecting chords of slope $m$ of $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ is:
- A
  $y=-\dfrac{b^{2}}{a^{2}m}x$
- B
  $y=\dfrac{b^{2}}{a^{2}m}x$
- C
  $y=mx$
- D
  $y=-\dfrac{a^{2}}{b^{2}m}x$
Conjugate diameters
$m_{1} m_{2} = - \frac{b ^{2}}{a ^{2}}$
Q1MCQConjugate diameters
For conjugate diameters $y=m_1x,\ y=m_2x$ of the ellipse:
- A
  $m_1m_2=-\dfrac{b^{2}}{a^{2}}$
- B
  $m_1m_2=-1$
- C
  $m_1m_2=\dfrac{b^{2}}{a^{2}}$
- D
  $m_1+m_2=0$
Co-normal points sum
$θ_{1} + θ_{2} + θ_{3} + θ_{4} = (2 n + 1) π$
eccentric angles
Q1MCQCo-normal points
The sum of the eccentric angles of the four co-normal points on an ellipse is:
- A
  an odd multiple of $\pi$
- B
  an even multiple of $\pi$
- C
  $0$
- D
  $\pi/2$

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Ellipse formulas

Ellipse, every formula

Standard equation

Eccentricity (a>b)

Eccentricity condition

Foci

Vertices

Directrices

Length of major & minor axes

Length of latus rectum

Ends of latus rectum

Sum of focal distances

Focal distances

Distance between foci

Parametric point

Notation S₁₁

Position of a point

Focal chord (semi-LR = HM)

Chord joining α,β

Tangent at (x₁,y₁)

Tangent at θ

Tangent of slope m

Tangency condition

Point of contact (slope form)

Normal at (x₁,y₁)

Normal at θ

Chord of contact

Director circle

Auxiliary circle

Area of ellipse

Max inscribed rectangle

Product of ⊥ from foci to tangent

Midpoint chord

Diameter (bisecting slope-m chords)

Conjugate diameters

Co-normal points sum

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