»Write down the equation of an ellipse, that goes through $\displaystyle P(1,-2)$ and whose circumscribed* rectangle's area is minimized.«

*

(Pretending not knowing that "rectangle" is in fact square.)

Set-up:

Ellipse through $\displaystyle (1,-2)$: $\displaystyle \frac{1}{a^2}+\frac{4}{b^2}=1$, rectangle's area: $\displaystyle 2a*2b=min.$

Now, what to differentiate? From the first equation I've also got $\displaystyle a^2b^2=b^2+4a^2$.