Chapters

Slope Formulas
Line Formulas
Line in Space
Plane Formulas
Conic Sections Formulas

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.

Analytic Geometry Formulas

Slope Formulas

Slope of two given points

Slope of the given angle

Slope of a given vector of the line

Slope of the given equation

Line Formulas

Distance between two points

Vector equation

Parametric form

Point slope

General form

Slope-Intercept form

Intercept form

Two-points form

Vertical and horizontal lines

Parallel lines

Perpendicular lines

Angle between two lines

Distance from a point to a line

Perpendicular bisector

Angle bisector

Line in Space

Vector form

Parametric form

Cartesian equations

Intercept form

Plane Formulas

Vectorial Equation of the plane

Parametric equations of the plane

Cartesian equation of the plane

Intercept form

Conic Sections Formulas

Circle Equations

Circle with the origin at its center

Ellipse

Eccentricity

Horizontal major axis

Vertical major axis

Hyperbola

Eccentricity

Asymptotes

Horizontal transverse axis

Vertical transverse axis

Rectangular hyperbola

Asymptotes

Eccentricity

Equation determined by the asymptotes

Parabola

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

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

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

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

Theory

Two-Point Form

Solved Line Word Problems

Point–Slope Form

Intersection Between Two Lines

Angle between Two Lines

Line-Plane Intersection

General Form

Normal Vector

Horizontal and Vertical Lines

Parallel Lines

Locus

Midpoint

Slope–Intercept Form

Intercept Form

Slope

Collinear Points

Vector Equation of a Line

Equation of Lines in Space

Parametric Form

Perpendicular Lines

Distance from a Point to a Line

Formulas

Line Formulas

Distance Between Two Points

Slope Formulas

Exercises