Ellipse formulas

Every formula JEE Advanced expects you to know for Ellipse, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 34
Subject: Mathematics
Exam: JEE Advanced
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Ellipse, every formula

34 formulas, typeset and free. Print it, or keep it open beside your practice.

Standard equation
$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$
Eccentricity (a>b)
$e = 1 - \frac{b ^{2}}{a ^{2}}, b^{2} = a^{2} (1 - e^{2})$
Eccentricity condition
$0 < e < 1$
Foci
$(\pm a e, 0)$
Vertices
$(\pm a, 0)$
Directrices
$x = \pm \frac{a}{e}$
Length of major & minor axes
$2 a (major), 2 b (minor)$
Length of latus rectum
$\frac{2 b ^{2}}{a}$
Ends of latus rectum
$(\pm a e, \pm \frac{b ^{2}}{a})$
Sum of focal distances
$S P + S^{'} P = 2 a$
Focal distances
$S P = a - e x_{1}, S^{'} P = a + e x_{1}$
Distance between foci
$S S^{'} = 2 a e$
Parametric point
$(a cos θ, b sin θ)$
Notation S₁₁
$S_{11} = \frac{x _{1}^{2}}{a ^{2}} + \frac{y _{1}^{2}}{b ^{2}} - 1$
Position of a point
$S_{11} < 0 inside, = 0 on, > 0 outside$
Focal chord (semi-LR = HM)
$\frac{1}{S P} + \frac{1}{S ^{'} P} = \frac{2}{S L}$
Chord joining α,β
$\frac{x}{a} cos \frac{α + β}{2} + \frac{y}{b} sin \frac{α + β}{2} = cos \frac{α - β}{2}$
Tangent at (x₁,y₁)
$\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1$
Tangent at θ
$\frac{x}{a} cos θ + \frac{y}{b} sin θ = 1$
Tangent of slope m
$y = m x \pm a^{2} m^{2} + b^{2}$
Tangency condition
$c^{2} = a^{2} m^{2} + b^{2}$
for y=mx+c
Point of contact (slope form)
$(- \frac{a ^{2} m}{c}, \frac{b ^{2}}{c})$
Normal at (x₁,y₁)
$\frac{a ^{2} x}{x _{1}} - \frac{b ^{2} y}{y _{1}} = a^{2} - b^{2}$
Normal at θ
$\frac{a x}{cos θ} - \frac{b y}{sin θ} = a^{2} - b^{2}$
Chord of contact
$\frac{x x _{1}}{a ^{2}} + \frac{y y _{1}}{b ^{2}} = 1 (S_{1} = 0)$
Director circle
$x^{2} + y^{2} = a^{2} + b^{2}$
Auxiliary circle
$x^{2} + y^{2} = a^{2}$
Area of ellipse
$π ab$
Max inscribed rectangle
$2 ab, sides a 2, b 2$
Product of ⊥ from foci to tangent
$b^{2}$
Midpoint chord
$S_{1} = S_{11}$
Diameter (bisecting slope-m chords)
$y = - \frac{b ^{2}}{a ^{2} m} x$
Conjugate diameters
$m_{1} m_{2} = - \frac{b ^{2}}{a ^{2}}$
Co-normal points sum
$θ_{1} + θ_{2} + θ_{3} + θ_{4} = (2 n + 1) π$
eccentric angles

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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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