Question

In the X-Y plane, the area of the region bounded by the graph of \(|x+y| + |x-y| = 4\) is

• A. 8
• B. 12
• C. 16
• D. 20

Hint:

For any two numbers a and b, \(|a| + |b| = \max(|a+b|, |a-b|)\). Apply this with a = x + y and b = x - y.

Answer 1

The Correct Option is C

Step 1: Simplify the equation:
Put \(a = x + y\), \(b = x - y\). Then \(|a| + |b| = \max(|a + b|, |a - b|) = \max(|2x|, |2y|) = 2\max(|x|, |y|)\).

Step 2: Identify the curve:
So the equation becomes \(\max(|x|, |y|) = 2\). This is the boundary of the square with vertices \((\pm 2, \pm 2)\), that is the lines \(x = \pm 2\) and \(y = \pm 2\).

Step 3: Find the area:
The side of the square is \(2 - (-2) = 4\), so the area is \(4 \times 4 = 16\).

Final Answer:
The correct option is (3): 16