Write the equation of the ellipse, knowing it’s center C ( 2 , 1 ) and the ends of two conjugate diameters A ( 5 , 1 ), B ( 0 , 3 ) .

Question

Write the equation of the ellipse, knowing it’s center   C  (  2  ,  1  ) and the ends of two conjugate diameters   A  (  5  ,  1  ),   B  (  0  ,  3  ) .

Korotnokby · Accepted Answer

Step 1The parametric equation of the ellipse using its conjugate radii                     C        A            →        ,                    C        B            →       is            P      →        (  t  )  =            C      →        +  cos  ⁡  t                    C        A            →        +  sin  ⁡  t                    C        B            →          (  1  )In matrix-vector form, let   P  =  [  x  ,  y      ]    T    ,  C  =  [  2  ,  1      ]    T   then   C  A  =  [  3  ,  0      ]    T   and   C  B  =  [  −  2  ,  2      ]    T   and   u  =  [  cos  ⁡  t  ,  sin  ⁡  t      ]    T   , the above equation becomes  P  −  C  =  G  u    (  2  )where   G  =      [                            3                                  −          2                                      0                                  2                      ]  From equation (2) we have  u  =      G          −      1        (  P  −  C  )    (  3  )And since       cos    2    ⁡  t  +      sin    2    ⁡  t  =  1 , then       u    T    u  =  1 , hence,  (  P  −  C      )    T        G          −      T            G          −      1        (  P  −  C  )  =  1    (  4  )And this is the equation of the ellipse. For numerical evaluation, we have      G          −      1        =            1      6            [                            2                                  2                                      0                                  3                      ]  Thus      G          −      T            G          −      1        =            1      36            [                            4                                  4                                      4                                  13                      ]  so that the equation of the ellipse (4) becomes  4  (  x  −  2      )    2    +  8  (  x  −  2  )  (  y  −  1  )  +  13  (  y  −  1      )    2    =  36

Write the equation of the ellipse, knowing it’s center C ( 2 , 1 ) and the ends

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