Question

The set \( \{ (x, y) : |x| + |y| = 1 \ \) in the \( xy \) plane represents:}

• A. a square
• B. a circle
• C. an ellipse
• D. a rectangle which is not a square
• E. a rhombus which is not a square

Hint:

This shape is often called a "unit ball" in the \( L_1 \) norm. Visually, it looks like a square tilted at 45 degrees, centered at the origin.

Answer 1

The Correct Option is A

Concept: The equation \( |x| + |y| = 1 \) involves absolute values, which means the relationship changes depending on the quadrant the point \( (x, y) \) is in. We analyze the equation in each of the four quadrants to determine the shape formed by the boundaries.

Step 1: Analyze the equation in each quadrant.

• Quadrant I (\( x \geq 0, y \geq 0 \)): The equation becomes \( x + y = 1 \). This is a line segment connecting \( (1, 0) \) and \( (0, 1) \).
• Quadrant II (\( x < 0, y \geq 0 \)): The equation becomes \( -x + y = 1 \). This is a line segment connecting \( (-1, 0) \) and \( (0, 1) \).
• Quadrant III (\( x < 0, y < 0 \)): The equation becomes \( -x - y = 1 \). This is a line segment connecting \( (-1, 0) \) and \( (0, -1) \).
• Quadrant IV (\( x \geq 0, y < 0 \)): The equation becomes \( x - y = 1 \). This is a line segment connecting \( (1, 0) \) and \( (0, -1) \).

Step 2: Determine the properties of the resulting shape.
The four line segments join to form a closed quadrilateral with vertices at \( (1, 0), (0, 1), (-1, 0), \) and \( (0, -1) \).
• The length of each side is \( \sqrt{(1-0)^2 + (0-1)^2} = \sqrt{2} \). Since all sides are equal, it is a rhombus.
• The slopes of adjacent sides are \( -1 \) (for \( x+y=1 \)) and \( 1 \) (for \( -x+y=1 \)). Since the product of slopes is \( -1 \), the sides are perpendicular. A rhombus with perpendicular sides (all angles are \( 90^\circ \)) is a square.