Question 1136328

given: ellipse with a focus at the pole and major axis endpoints ({{{4}}}, {{{0}}}) and ({{{2}}}, {{{pi}}}).


{{{r = ke / ( 1 - e *cos (theta))}}}

 


substitute in points:


{{{4 = ke/ ( 1 - cos (0) )}}}==> {{{cos (0)=1}}}


{{{4 = ke/ ( 1 - e(1) )}}}


{{{4 = ke / ( 1 - e ) }}}


{{{ke=4( 1 - e ) }}}


{{{ke=4 - 4e}}}


{{{k=4/e-4}}}......eq.1



{{{2 = ke/ ( 1 -cos (pi))}}}==> {{{cos (pi)=-1}}}


{{{2 = ke/ ( 1 - e(-1) )}}}


{{{2 = ke/ ( 1 + e )}}}


 {{{ke=2 + 2e }}}
 

{{{k=2/e+2}}}......eq.2



from eq.1 and eq.2 we have


{{{4/e-4=2/e+2}}}


{{{4/e-2/e=2+4}}}


{{{2/e=6}}}


{{{2=6*e}}}


{{{e=2/6}}}


{{{e=1/3}}}



find {{{k}}}


{{{k=4/e-4}}}......eq.1


{{{k=4/(1/3)-4}}}


{{{k=12-4}}}


{{{k=8}}}



{{{r = ke / ( 1 - e *cos (theta))}}}...substitute in {{{k}}} and {{{e}}}


{{{r =(8(1/3)) / ( 1 - (1/3) *cos (theta))}}}


{{{r =(8/3) / ( 1 - (1/3) *cos (theta))}}}......both numerator and denominator multiply by {{{3}}}


{{{r =3(8/3) / ( 3 - 3(1/3) *cos (theta))}}}


{{{r =8 / ( 3 - cos (theta))}}}