Latus rectum

Let L and L' be the ends of the latus rectum of the parabola y2=4ax. Since S = (a,0) , the x-cocrdinates of L and L' are also equal to 'a'. Let L = (a,b). Since L is a point of the parabola, we have:
Latus rectum
b2 = 4a( a) = 4a2
b = ±2a
L = (a,2a) ,L’ = (a,-2a ) and
the length of the latus rectum LL’ is 4a.

 

Length of Latus rectum

 

Length of latus rectum of hyperbola:

Latus rectum of hyperbola is defined analogously as in the case of parabola and ellipse.Therefore for Hyperbola,

The ends of latus rectum are  (ae, ±b2/a2)  and the length of latus rectum is 2b2/a

 

Latus rectum Summary

 

             Conic Section                       Length of latus rectum                         Ends of latus rectum

                  y2 = 4ax                                              4a                                               L = (a,2a) , L = (a,-2a)  

    ( x2/a2 ) + (y2/b2)  = 1 ; a>b                          2b2/a                                            L = (ae,b2/a) , L’ = (ae,-b2/a)

     ( x2/a2)  + (y2/b2) = 1 ; b>a                          2a2/b                                            L = (ae,b2/a) , L’ = (ae,-b2/a)

     ( x2/a2 ) - (y2/b2)  = 1 ; a>b                          2a2/b                                            L = (ae,b2/a) , L’ = (ae,-b2/a)

 

Solved problems

 

1. Find the length of latus rectum of parabola, y2 = 12x.

Solution:          y2 = 12x
                    => y2 = 4(3)x   Since y2 = 4ax is the  equation of parabola,
                           a = 3
            Therefore , length of latus rectum = 4a = 4(3) =12.

 

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2. Find the length of latus rectum of ellipse 4x2 + 9y2 - 24x + 36y  - 72 = 0.

Solution: 4x2 + 9y2 - 24x + 36y  - 72 = 0
           => ( 4x2 - 24x ) + ( 9y2 + 36y ) - 72 = 0
           => 4( x2 -6x ) + 9( y2 + 4y ) - 72 = 0
           => 4[ x2 - 6x +9 ] + 9[ y2 + 4y +4 ] = 144
           => 4( x - 3 )2 + 9( y + 2 )2 = 144
           => {( x - 3 )2/ 36} + {( y + 2 )2/ 16 } = 1
           => {( x - 3 )2/ 62} + {( y + 2 )2/ 42 } = 1
           => a = 3  and  b = 2

Therefore , length of latus rectum = 2b2/a = 2(4)2/6 = 16/3.