Hello, shenton!
This is JakeD's solution ... with a diagram.
What is the area of a square inscribed in an ellipse 16x² + 9y² = 144? Code:

***
B *  * A
(p,q)*+*(p,q)
*:  :*
:  :
* :  : *
   * :    +    : *   
* :  : *
:  :
*:  :*
*+*(p,q)
*  * C
***

We want points A, B, C on the ellipse such that: .AB = AC.
. . That is: .2p = 2q . → . p = q
. . . . . . . . . . . . . . . . . . . . . . . . . . . . _____
Solve the equation for y: . y .= .±(4/3)√9  x²
. . . . . . . . . . . .   . . . . . . . . . . . ._____
Then p = q becomes: .p .= .±(4/3)√9  p²
. . and we have: .9p² .= .16(9  p²) . → . x² .= .144/25
Hence: .p = 12/5 = 2.4 .and the side of the square is 4.8 units.
Therefore, the area of the square is: .4.8² = 23.04 units².