Hello, cinematic!
Your work is correct . . . up to the punchline.
. . Did you forget to "double"?
Find the dimensions of the largest rectangle with sides parallel to the axes
that can be inscribed in the ellipse: Code:

* * *
*  *
o      +      o
*:  :*
:  y:
* :  : *
  * :      +      : *  
* :  x : *
:  :
*:  :*
o      +      o
*  *
* * *

The upper half of the ellipse is: .
The area of the rectangle is: .
We have: .
Then: .
. . .
. . .
Hence: .
Therefore: .