Parabolas and Ellipses

Conic form for a vertical parabola

(x-h)²=4p(y-k)

Conic form for a horizontal parabola

(y-k)²=4p(x-h)

What does p represent?

The distance from the vertex to the focus OR the distance from the vertex to the directrix

Axis of symmetry for a vertical parabola

x=h

Axis of symmetry for a horizontal parabola

y=k

Focus for a vertical parabola

(h,k+p)

Focus for a horizontal parabola

(h+p,k)

Directrix for a vertical parabola

y=k-p

Directrix for a horizontal parabola

x=h-p

Length of Latus Rectum

|4p|

Conic form for a horizontal ellipse

(x-h)²/a²+(y-k)²/b²=1

Conic form for a vertical ellipse

(x-h)²/b²+(y-k)²/a²=1

c²=?

c²=a²-b²

Vertices of a horizontal ellipse

(h±a,k)

Vertices of a vertical ellipse

(h,k±a)

Co-vertices of horizontal ellipse

(h,k±b)

Co-vertices of vertical ellipse

(h±b,k)

Major axis of horizontal ellipse

y=k

Minor axis of horizontal ellipse

x=h

Major axis of vertical ellipse

x=h

Minor axis of vertical ellipse

y=k

Foci of horizontal ellipse

(h±c,k)

Foci of vertical ellipse

(h,k±c)

Major axis Length

2a

Minor Axis Length

2b

Eccentricity

e=c/a (0<e<1)