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:

y−y0y−y0m1m2=m1(x−x0)=m2(x−x0)=y0+y20−2px0−−−−−−−−√2x0=y0−y20−2px0−−−−−−−−√2x0

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=a2−b2−−−−−−√a

Foci of the ellipse:

if a≥b⟹F1(−a2−b2−−−−−−√,0)  F2(a2−b2−−−−−−√,0)if a<b⟹F1(0,−b2−a2−−−−−−√)  F2(0,b2−a2−−−−−−√)

Area of the ellipse:

A=π⋅a⋅b

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:

x2a2−y2b2=1

Parametric equations of the Hyperbola:

xy=asint=bsintcost

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

x0xa2−y0yb2=1

Foci:

if a≥b⟹F1(−a2+b2−−−−−−√,0)  F2(a2+b2−−−−−−√,0)if a<b⟹F1(0,−a2+b2−−−−−−√)  F2(0,a2+b2−−−−−−√)

Asymptotes:

if a≥b⟹y=bax and y=−baxif a<b⟹y=abx and y=−abx