Question

Area of the region which consists of all the points satisfying the conditions |x − y| + |x + y| ≤ 8 and xy≥ 2, is equal to

• A. 4(7 – ln 8) sq. units
• B. 4(9 −ln 8) sq. units
• C. 2(7 – ln 8) sq. units
• D. 2(9 – ln8) sq. union

Answer 1

Answer: (A)

4(7 – ln 8) sq. units

Answer 2

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,

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

= 4(7 − 3 ln2) sq. units.

= 4 (7 − ln8) sq. units.