Area of the region which consists of all the points satisfying the conditions $|x-y|+|x+y| \leq 8$ and $x y \geq 2$, is equal to

4(7 – ln 8) sq. units

The expression |x − y| + |x + y| ≤ 8, represents the interior region of the square formed by the lines x = ±4, y = ±4 and xy ≥ 2.

Represents the region lying inside the hyperbola xy = 2

Required area

$\Delta=2 \int\limits_{1 / 2}^4(\left(4-\frac{2}{x}\right) d x=2(4 x-2 l x)_{1 / 2}^4$

= 4(7 - 3 ln 2) sq. units.

= 4(7 - ln 8) sq. units.