Math formulas and cheat sheets for Conic Sections

Conic Sections

The Parabola Formulas

The standard formula of a parabola

y2=2px

Parametric equations of the parabola:

xy=2pt2=2pt

Tangent line in a point D(x0,y0) of a parabola y2=2px is :

y0y=p(x+x0)

Tangent line with a given slope m:

y=mx+p2m

Tangent lines from a given point

Take a fixed point P(x0,y0). The equations of the tangent lines are:

yy0yy0m1m2=m1(xx0)=m2(xx0)=y0+y202px02x0=y0y202px02x0

The Ellipse Formulas

The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant.

The standard formula of a ellipse:

x2a2+y2b2=1

Parametric equations of the ellipse:

xy=acost=bsint

Tangent line in a point D(x0,y0) of a ellipse:

x0xa2+y0yb2=1

Eccentricity of the ellipse:

e=a2b2a

Foci of the ellipse:

if abF1(a2b2,0)  F2(a2b2,0)if a<bF1(0,b2a2)  F2(0,b2a2)

Area of the ellipse:

A=πab

The Hyperbola Formulas

The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

The standard formula of a hyperbola:

x2a2y2b2=1

Parametric equations of the Hyperbola:

xy=asint=bsintcost

Tangent line in a point D(x0,y0) of a Hyperbola:

x0xa2y0yb2=1

Foci:

if abF1(a2+b2,0)  F2(a2+b2,0)if a<bF1(0,a2+b2)  F2(0,a2+b2)

Asymptotes:

if aby=bax and y=baxif a<by=abx and y=abx