The area of a quadrilateral formed with the focii of the conics x 2 a 2 + y 2 b 2 = 1 and x 2 a 2 − y 2 b 2 = − 1 is?The points of the quadrilateral are ( ± a e 1 , 0 ) and ( 0 , ± b e 2 )The area of half of the quadrilateral (a triangle) is Δ = 1 2 ( 2 a e 1 ) ( b e 2 ) Δ = a b e 1 e 2 Also e 1 = 1 − b 2 a 2 e 2 = 1 + a 2 b 2 Therefore Δ = a b a 4 − b 4 a 2 b 2 Area of quadrilateral is = 2 a 4 − b 4 .But the answer is 2 ( a 2 + b 2 ). Where am I going wrong?

Question

The area of a quadrilateral formed with the focii of the conics             x      2              a      2        +            y      2              b      2        =  1 and             x      2              a      2        −            y      2              b      2        =  −  1 is?The points of the quadrilateral are   (  ±  a      e    1    ,  0  ) and   (  0  ,  ±  b      e    2    )The area of half of the quadrilateral (a triangle) is   Δ  =      1    2    (  2  a      e    1    )  (  b      e    2    )  Δ  =  a  b      e    1        e    2  Also       e    1    =      1    −                  b        2                    a        2                  e    2    =      1    +                  a        2                    b        2            Therefore   Δ  =  a  b                              a          4                −                  b          4                                      a          2                          b          2                    Area of quadrilateral is   =  2            a      4        −          b      4      .But the answer is   2  (      a    2    +      b    2    ). Where am I going wrong?

Sanai Douglas · Accepted Answer

Explanation:The area of the wuadeilater if   a  &amp;gt;  b needs to be   Q  =      1    2    (  2  a      e    1    )  (  2  b      e    2    ). So the area of the quadrilateral is  Q  =  2  a  b      e    1        e    2    =  2            a      4        −          b      4           i  f        a  &amp;gt;  bStep 2But if   a  &amp;lt;  b, all four foci will be collinear so no quadrilateral is formed.

The area of a quadrilateral formed with the focii of the conics x^2/a^2+y^2/b^2=1 and x^2/a^2-y^2/b^2=-1 is?

Answered question

Answer & Explanation

New Questions in High school geometry