[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part  [

An inscribed angle that subtends a diameter. Pythagorean Theorem (direct and inverse).

A circle passes through the adjacent vertices $M$ and $N$ of the rectangle $M N P Q$. The length of the tangent drawn from point $Q$ to the circle is $1, P Q=2$. Find all possible values that the area of the rectangle $M N P Q$ can take if the diameter of the circle is $\sqrt{5}$.
Let $Q M=x$, and the line $Q M$ intersects the circle at point $L$. From the right triangle $L M N$, we find that $L M=1$. According to the tangent-secant theorem, $Q M \cdot Q L=1$, that is, $x(x \pm 1)=1$ (the sign depends on whether point $L$ lies on the segment $Q M$ or outside it). Solving this equation, we get $S_{M N P Q}=2 x=\sqrt{5} \pm 1$.

## Answer

$\sqrt{5} \pm 1$.