Question 925977
latus rectum of an ellipse  ={{{ 2b^2/a}}}
 eccentricity  =  {{{ sqrt(1-b^2/a^2)}}}
{{{ 2b^2/a  =5}}}
   move a to the right
  2b^2 = 5a
 divide with 2 on both sides
 {{{ 2b^2/2 =  5a/2}}}
    {{{ b^2 =5a/2}}}
put above in eccentricity equation
 
 eccentricity  =  {{{ sqrt(1-b^2/a^2)}}}
     {{{2/3 = sqrt(1-b^2/a^2)}}}
     {{{2/3 =sqrt(1-(5a/2)/a^2)}}}
 {{{2/3 =sqrt(1-5/2a)}}} 

squaring on both sides
 {{{ (2/3)^2  =(sqrt(1-5/2a))^2}}} 
      {{{ 4/9=1-5/2a}}}
     move 1 to the left
     {{{4/9-1 =-5/2a}}}
      {{{ 4/9 -9/9 =-5/2a}}}
       {{{ (4-9)/9 =-5/2a}}}
              {{{-5/9 =-5/2a}}}
                {{{5/9 =5/2a}}}
               divide with 5 on both sides
       {{{ 5/9 *1/5 = 5/2a*1/5}}}
             {{{ 1/9 = 1/2a}}}
              2a  =9
  on dividing with 2 on both sides
       a  =  9/2
but {{{ b^2  = 5*a/2}}}
             ={{{5* (9/2)/2}}}
             ={{{45/4}}}
     {{{  b = sqrt(45/4)}}}