Slope Formulas

Slope of two given points

m = \frac { { y }_{ 2 } - { y }_{ 1 } }{ { x }_{ 2 } - { x }_{ 1 } }

Slope of the given angle

m = \tan { \alpha }

Slope of a given vector of the line

m = \frac { { v }_{ 2 } }{ { v }_{ 2 } }

Slope of the given equation

Compare the equation with y = mx + c, where m is the gradient of that specific line.

 

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Line Formulas

Distance between two points

A({ x }_{ 1 }, { y }_{ 1 }) \qquad B({ x }_{ 2 }, { y }_{ 2 })

d(A, B) = \sqrt { { { x }_{ 2 } - { x }_{ 1 } }^{ 2 } + { { y }_{ 2 } - { y }_{ 1 } }^{ 2 } }

Vector equation

(x, y) = ({ x }_{ 1 }, { y }_{ 1 }) + t . ({ v }_{ 1 }, { v }_{ 2 })

Parametric form

\begin{cases} x = { x }_{ 1 } + t . { v }_{ 1 } \\ y = { y }_{ 1 } + t . { v }_{ 2 } \end{cases}

Point slope

y - { y }_{ 1 } = m(x - { x }_{ 1 })

General form

Ax + By + C = 0

\overrightarrow { v } = (-B , A) \qquad m = - \frac { A }{ B }

Slope-Intercept form

y = mx + c, where c is the y-intercept

Intercept form

\frac { x }{ a } + \frac { y }{ b } = 1

Two-points form

\frac { x - { x }_{ 1 } }{ { x }_{ 2 } - { x }_{ 1 } } = \frac { y - { y }_{ 1 } }{ { y }_{ 2 } - { y }_{ 1 } }

Vertical and horizontal lines

y = 0x + b \qquad y = b

x = a

Parallel lines

\overrightarrow { u }  = \overrightarrow { v }

\frac { { u }_{ 1 } }{ { u }_{ 2 } } = \frac { { v }_{ 1 } }{ { v }_{ 2 } } \qquad \frac { { A }_{ 1 } }{ { B }_{ 1 } } = \frac { { A }_{ 2 } }{ { B }_{ 2 } }

{ m }_{ r } = { m }_{ s }

Perpendicular lines

{ m }_{ r } = - \frac { 1 }{ { m }_{ s } }

Angle between two lines

\cos { \alpha } = \frac { \left| { u }_{ 1 } . { v }_{ 1 } + { u }_{ 2 } . { v }_{ 2 } \right| }{ \sqrt { { { u }_{ 1 } }^{ 2 } + { u }_{ 2 } }^{ 2 } } . \sqrt { { { v }_{ 1 } }^{ 2 } + { v }_{ 2 } }^{ 2 } }  }

\tan { \alpha } = \left| \frac { { m }_{ 2 } - { m }_{ 1 } }{ 1 + { m }_{ 2 } . { m }_{ 1 } } \right|

Distance from a point to a line

d(P, r) = \frac { \left| A . { p }_{ 1 } + B . { p }_{ 2 } + C \right| }{ \sqrt { { A }^{ 2 } + { B }^{ 2 } } }

Perpendicular bisector

\sqrt { { (x - { x }_{ 1 }) }^{ 2 } + { (y - { y }_{ 1 }) }^{ 2 } } = \sqrt { { (x - { x }_{ 2 }) }^{ 2 } + { (y - { y }_{ 2 }) }^{ 2 } }

Angle bisector

\frac { \left| { A }_{ 1 }x + { B }_{ 1 }y + { C }_{ 1 } \right| }{ \sqrt { { ( { A }_{ 1 } ) }^{ 2 } + { ( { B }_{ 2 } ) }^{ 2 } } }

 

Line in Space

Vector form

(x, y, z) = ( { x }_{ 0 }, { y }_{ 0 }, { z }_{ 0 } ) + \lambda . ( { u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 } )

Parametric form

\begin{cases} x = { x }_{ 0 } + \lambda . { u }_{ 1 } \\ y = { y }_{ 0 } + \lambda . { u }_{ 2 } \\ z = { z }_{ 0 } + \lambda . { u }_{ 3 } \end{cases}

Cartesian equations

\frac { x - { x }_{ 0 } }{ { u }_{ 1 } } = \frac { y - { y }_{ 0 } }{ { u }_{ 2 } } = \frac { z - { z }_{ 0 } }{ { u }_{ 3 } }

Intercept form

\frac { x }{ a } + \frac { y }{ b } + \frac { z }{ c } = 1

 

Plane Formulas

Vectorial Equation of the plane

\overrightarrow { PX } = \lambda \overrightarrow { u } + \mu \overrightarrow { v }

(x, y, z) = ({ x }_{ 0 }, { y }_{ 0 }, { z }_{ 0 }) + \lambda ({ u }_{ 1 }, { u }_{ 2 }, { u }_{ 3 }) + \mu ({ v }_{ 1 }, { v }_{ 2 }, { v }_{ 3 })

Parametric equations of the plane

\begin{cases} x = { x }_{ 0 } + \lambda . { u }_{ 1 } + \mu . { v }_{ 1 } \\ y = { y }_{ 0 } + \lambda . { u }_{ 2 } + \mu . { v }_{ 2 } \\ z = { z }_{ 0 } + \lambda . { u }_{ 3 } + \mu . { v }_{ 3 } \end{cases}

Cartesian equation of the plane

\begin{vmatrix} x - { x }_{ 0 } & { u }_{ 1 } & { v }_{ 1 } \\ y - { y }_{ 0 } & { u }_{ 2 } & { v }_{ 2 } \\ z - { z }_{ 0 } & { u }_{ 3 } & { v }_{ 3 } \end{vmatrix} = 0

Ax + By + Cz + D = 0

Intercept form

\frac { x }{ a } + \frac { y }{ b } + \frac { z }{ c } = 1

a = \frac { -D }{ A } \qquad b = \frac { -D }{ B } \qquad c = \frac { -D }{ C }

 

Conic Sections Formulas

Circle Equations

{ (x - a) }^{ 2 } + { (y - b) }^{ 2 } = { r }^{ 2 }

{ x }^{ 2 } + { y }^{ 2 } + Ax + By + C = 0

A = -2a \qquad B = -2b \qquad C = { a }^{ 2 } + { b }^{ 2 } - { r }^{ 2 }

C = (- \frac { A }{ 2 }, - \frac { B }{ 2 } ) { r }^{ 2 } = { (\frac { A }{ 2 }) }^{ 2 } + { (\frac { B }{ 2 }) }^{ 2 } - C

Circle with the origin at its center

{ x }^{ 2 } + { y }^{ 2 } = { r }^{ 2 }

Ellipse

\overline { PF } + \overline { PF' } = 2a

{ a }^{ 2 } = { b }^{ 2 } + { c }^{ 2 }

Eccentricity

e = \frac { c }{ a } \qquad c \le a \qquad 0 \le e \le 1

Horizontal major axis

\frac { { x }^{ 2 } }{ { a }^{ 2 } } + \frac { { y }^{ 2 } }{ { b }^{ 2 } } = \frac { { { (x-{ x }_{ 0 }) }^{ 2 } } }{ { a }^{ 2 } } +\frac { ({ y-{ y }_{ 0 }) }^{ 2 } }{ { b }^{ 2 } } = 1

Vertical major axis

\frac { { y }^{ 2 } }{ { a }^{ 2 } } + \frac { { x }^{ 2 } }{ { b }^{ 2 } } = \frac { { { (y-{ y }_{ 0 }) }^{ 2 } } }{ { a }^{ 2 } } +\frac { ({ x-{ x }_{ 0 }) }^{ 2 } }{ { b }^{ 2 } } = 1

Hyperbola

\overline { PF } - \overline { PF' } = 2a

{ a }^{ 2 } = { b }^{ 2 } + { c }^{ 2 }

Eccentricity

e = \frac { c }{ a } \qquad c \le a \qquad e \ge 1

Asymptotes

y = - \frac { b }{ a }x, \qquad y = \frac { b }{ a }x

Horizontal transverse axis

\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =\frac { { { (x-{ x }_{ 0 }) }^{ 2 } } }{ { a }^{ 2 } } -\frac { ({ y-{ y }_{ 0 }) }^{ 2 } }{ { b }^{ 2 } } =1

Vertical transverse axis

\frac { { y }^{ 2 } }{ { a }^{ 2 } } -\frac { { x }^{ 2 } }{ { b }^{ 2 } } =\frac { { { (y-{ y }_{ 0 }) }^{ 2 } } }{ { a }^{ 2 } } -\frac { ({ x-{ x }_{ 0 }) }^{ 2 } }{ { b }^{ 2 } } =1

Rectangular hyperbola

{ x }^{ 2 } - { y }^{ 2 } = { a }^{ 2 }

Asymptotes

y = x, \qquad y = -x

Eccentricity

e = \sqrt { 2 }

Equation determined by the asymptotes

x . y = \frac { { a }^{ 2 } }{ 2 } \qquad x . y = k

Parabola

Parabolas with vertex at (0,0) and axis on the y-axis

F = (0, - \frac { p }{ 2 }) \qquad y = - \frac { p }{ 2 }

{ x }^{ 2 } = 2py

F = (0, - \frac { p }{ 2 }) \qquad y = \frac { p }{ 2 }

{ x }^{ 2 } = -2py

Parabola with vertex at (a,b) and axis parallel to the y-axis

F = (a, b + \frac { p }{ 2 }) \qquad y = b - \frac { p }{ 2 } \qquad V(a,b)

{ (x - a) }^{ 2 } = 2p(y - b)

Parabola with vertex at (0,0) and axis on the x-axis

F = (\frac { p }{ 2 } , 0) \qquad x = - \frac { p }{ 2 }

{ y }^{ 2 } = 2px

F = (- \frac { p }{ 2 }, 0) \qquad x = \frac { p }{ 2 }

{ y }^{ 2 } = -2px

Parabola with vertex at (a,b) and axis parallel to the x-axis

F = (a + \frac { p }{ 2 }, b ) \qquad x = a - \frac { p }{ 2 } \qquad V(a,b)

{ (y - b) }^{ 2 } = 2p(x - a)

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.