2.26. Cartesian Equations For The Parabola

Consider the conics equation for a parabola

        XF =| ( XP ) N ˆ |                     (2.21)

Taking X=( x,y ) , N ˆ =( 1,0 ) , F=( c,0 )  and the directrix a vertical line at x=α , (2.21) becomes

        ( xc ) 2 + y 2 =| xα |

Þ            ( xc ) 2 + y 2 = ( xα ) 2

or             y 2 =2( cα )x+ α 2 c 2          (2.43a)

The standard form of a parabola is obtained by setting α=c  so that (2.43a) becomes

        y 2 =4cx                   (2.43)

where the vertex is at the origin.  A parabola with vertex at ( x 0 , y 0 )  is simply

        ( y y 0 ) 2 =4c( x x 0 )

Note that a parbola does not have any asymptotes.