2.25. Cartesian Equations For The Ellipse And The Hyperbola in Standard Position

Consider again the central conics

        X 2 e 2 ( X N ˆ ) 2 = a 2 ( 1 e 2 )                 (2.37)

Setting X=( x,y )  and N ˆ =( 1,0 ) , we have

        x 2 + y 2 e 2 x 2 = a 2 ( 1 e 2 )

Þ            x 2 a 2 + y 2 a 2 ( 1 e 2 ) =1                           (2.38)

which is called the standard form of the central conics.  The foci are at ±ae N ˆ =( ±ae,0 ) .  The directrices are vertical lines intersecting the x-axis at points ± a e N ˆ =( ± a e ,0 )

For an ellipse with e<1 , we set b=a 1 e 2  so that (2.38) becomes

        x 2 a 2 + y 2 b 2 =1                                               (2.39)

and

        c=ae   =a 1 b 2 a 2   = a 2 b 2

For a hyperbola with e>1 , we set b=| a | e 2 1  so that (2.38) becomes

        x 2 a 2 y 2 b 2 =1                                               (2.40)

and

        c=| a |e   =| a | 1+ b 2 a 2   = a 2 + b 2

Now, as | x | , (2.40) becomes

        y 2 b 2 x 2 a 2            or             y± b | a | x

These give 2 lines with tangents ± b | a |  that pass through the origin.  They are called asymptotes of the hyperbola.  See Fig.2.14.