2.21. Polar Equations For Conic Sections

Setting the fous at the origin, i.e., F=0 , eq(2.22) simplifies to

        X     =e| X N ˆ d |                   (2.23)

which, in polar coordinates, becomes

        r=e| rcosθd |                       (2.24)

where r=     X  and X N ˆ =rcosθ  [see Fig.2.13].

 

If X is on the same side of the directrix as the focus, rcosθd<0  so that (2.24) becomes

        r=e( drcosθ )      Þ  r= ed ecosθ+1         (2.25)

 

If X is on the other side of the directrix from the focus, rcosθd>0  and (2.24) becomes

        r=e( rcosθd )      Þ  r= ed ecosθ1         (2.26)

Since, here, q is restricted to 0<θ< π 2  while r,e,d>0 , eq(2.26) cannot be satisfied for any q if e1 .  In other word, only hyperbola can have a branch on the other side of the directrix from the focus.

 

Consider now the branch indicated by (2.25).

If e<1 , there is no restriction on q so that 0<θ<2π  as befits an ellipse.

For e>1 , q is restricted to θ cos 1 ( 1 e ) .