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)$D(x_0, y_0)$ of a parabola y2=2px$y^2 = 2px$ is :

 y0y=p(x+x0)

Tangent line with a given slope m$m$:

 y=mx+p2m

Tangent lines from a given point

Take a fixed point P(x0,y0)$P(x_0, y_0)$. 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)$D(x_0, y_0)$ 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

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)$D(x_0, y_0)$ of a Hyperbola:

 x0xa2−y0yb2=1

Foci:

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

Asymptotes:

 if a≥b⟹y=bax and y=−baxif a