Analytic Geometry Formulas

A reference of analytic geometry formulas, grouped by category below. Every formula is typeset with KaTeX for clean, readable notation you can view online — click "Print" on any category to print it, or save it as a PDF straight from your browser's print dialog. Looking for a different subject? Browse the full list of math formula references.

Points and Lines

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

distance between two points

$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

midpoint of two points

$m=\frac{y_2-y_1}{x_2-x_1}$

slope between two points

$y=mx+b$

slope-intercept form of a line

$y-y_1=m(x-x_1)$

point-slope form of a line

$Ax+By=C$

standard form of a line

$m_1=m_2$

condition for two lines to be parallel

$m_1\cdot m_2=-1$

condition for two lines to be perpendicular

Circles

$(x-h)^2+(y-k)^2=r^2$

standard form — center $(h,k)$, radius $r$

$x^2+y^2+Dx+Ey+F=0$

general form

Parabolas

$(x-h)^2=4p(y-k)$

vertical axis — vertex $(h,k)$, focus at distance $p$

$(y-k)^2=4p(x-h)$

horizontal axis — vertex $(h,k)$, focus at distance $p$

Ellipses

$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$

center $(h,k)$, semi-axes $a$ and $b$

$e=\frac{c}{a}$

eccentricity — where $c^2=a^2-b^2$

Hyperbolas

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$

horizontal transverse axis

$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$

vertical transverse axis

$y-k=\pm\frac{b}{a}(x-h)$

asymptotes, for a horizontal transverse axis