Parabola formulas

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

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Formulas: 36
Subject: Mathematics
Exam: JEE Advanced
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Parabola, every formula

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

Conic eccentricity
$\frac{S P}{P M} = e$
e=1 parabola, <1 ellipse, >1 hyperbola
Parabola eccentricity
$e = 1$
Standard parabola
$y^{2} = 4 a x$
opens right, a>0
Vertex, focus
$A = (0, 0), S = (a, 0)$
for y²=4ax
Directrix & axis
$x = - a (directrix), y = 0 (axis)$
Length of latus rectum
$L L^{'} = 4 a$
Ends of latus rectum
$(a, \pm 2 a)$
Focal distance
$S P = x_{1} + a$
for y²=4ax
Parametric point
$(a t^{2}, 2 a t)$
on y²=4ax
Vertex form
$(y - k)^{2} = 4 a (x - h)$
vertex (h,k)
Axis parallel to x-axis
$x = a y^{2} + b y + c$
Notation S₁, S₁₁
$S_{1} \equiv y y_{1} - 2 a (x + x_{1}), S_{11} \equiv y_{1}^{2} - 4 a x_{1}$
Position of a point
$S_{11} < 0 inside, = 0 on, > 0 outside$
Chord joining t₁,t₂
$y (t_{1} + t_{2}) = 2 x + 2 a t_{1} t_{2}$
Focal chord condition
$t_{1} t_{2} = - 1$
Length of focal chord
$a (t + \frac{1}{t})^{2}$
Minimum focal chord (LR)
$least length = 4 a$
Semi-latus rectum (HM)
$\frac{1}{S P} + \frac{1}{S Q} = \frac{2}{l} = \frac{1}{a}$
Tangent at (x₁,y₁)
$y y_{1} = 2 a (x + x_{1})$
Tangent at parameter t
$y t = x + a t^{2}$
Tangent of slope m
$y = m x + \frac{a}{m}$
Condition y=mx+c tangent
$c = \frac{a}{m}$
Point of contact (slope form)
$(\frac{a}{m ^{2}}, \frac{2 a}{m})$
Normal at (x₁,y₁)
$y - y_{1} = - \frac{y _{1}}{2 a} (x - x_{1})$
Normal at parameter t
$y + x t = 2 a t + a t^{3}$
Normal of slope m
$y = m x - 2 am - a m^{3}$
Foot of normal (slope m)
$(a m^{2}, - 2 am)$
Perpendicular tangents locus
$x + a = 0 (directrix)$
Chord of contact
$y y_{1} = 2 a (x + x_{1}) (S_{1} = 0)$
Tangents from external point
$m^{2} x_{1} - m y_{1} + a = 0$
slopes m1,m2
Sum/product of tangent slopes
$m_{1} + m_{2} = \frac{y _{1}}{x _{1}}, m_{1} m_{2} = \frac{a}{x _{1}}$
Chord with midpoint
$S_{1} = S_{11} : y y_{1} - 2 a (x + x_{1}) = y_{1}^{2} - 2 a x_{1}$
Reflection property
$ray parallel to axis reflects through the focus$
Angle between tangents
$tan θ = \frac{S _{11}}{x _{1} + a}$
Normal chord: t₂ from t₁
$t_{2} = - t_{1} - \frac{2}{t _{1}}$
Circle on focal chord
$touches the directrix$

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Parabola formulas

Parabola, every formula

Conic eccentricity

Parabola eccentricity

Standard parabola

Vertex, focus

Directrix & axis

Length of latus rectum

Ends of latus rectum

Focal distance

Parametric point

Vertex form

Axis parallel to x-axis

Notation S₁, S₁₁

Position of a point

Chord joining t₁,t₂

Focal chord condition

Length of focal chord

Minimum focal chord (LR)

Semi-latus rectum (HM)

Tangent at (x₁,y₁)

Tangent at parameter t

Tangent of slope m

Condition y=mx+c tangent

Point of contact (slope form)

Normal at (x₁,y₁)

Normal at parameter t

Normal of slope m

Foot of normal (slope m)

Perpendicular tangents locus

Chord of contact

Tangents from external point

Sum/product of tangent slopes

Chord with midpoint

Reflection property

Angle between tangents

Normal chord: t₂ from t₁

Circle on focal chord

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