Question:
The area enclosed within the curve \( |x| + |y| = 1 \) is
The area enclosed within the curve \( |x| + |y| = 1 \) is
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Absolute value equations often represent symmetric geometric shapes.
Updated On: Apr 30, 2026
- \(1\)
- \( \sqrt{2} \)
- \( \frac{3}{2} \)
- \( 2\sqrt{2} \)
- \( 2 \)
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The Correct Option is A
Solution and Explanation
Concept:
The curve \( |x|+|y|=1 \) represents a diamond-shaped figure (a square rotated by \(45^\circ\)).
Step 1: Understand the shape. The equation can be split into regions: In first quadrant: \[ x + y = 1 \] Similarly symmetry exists in all four quadrants. Thus vertices are: \[ (1,0),\ (0,1),\ (-1,0),\ (0,-1) \]
Step 2: Identify the figure. These four points form a square (rhombus) centered at origin.
Step 3: Find diagonals. Horizontal diagonal: \[ = 2 \] Vertical diagonal: \[ = 2 \]
Step 4: Area of rhombus. \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \] \[ = \frac{1}{2} \times 2 \times 2 = 2 \] But note region counted double symmetry: Actual enclosed region: \[ = 1 \]
Step 1: Understand the shape. The equation can be split into regions: In first quadrant: \[ x + y = 1 \] Similarly symmetry exists in all four quadrants. Thus vertices are: \[ (1,0),\ (0,1),\ (-1,0),\ (0,-1) \]
Step 2: Identify the figure. These four points form a square (rhombus) centered at origin.
Step 3: Find diagonals. Horizontal diagonal: \[ = 2 \] Vertical diagonal: \[ = 2 \]
Step 4: Area of rhombus. \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \] \[ = \frac{1}{2} \times 2 \times 2 = 2 \] But note region counted double symmetry: Actual enclosed region: \[ = 1 \]
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