If the distance of any point `(x,y)` from origin is defined as `d(x,y)=max{|x|,|y|}`, then the locus of the point `(x,y)` where `d(x,y)=1` is

Options

A. a circle
B. a straight line
C. a square
D. a triangle

Answer

To find the locus of the point \((x,y)\) where the distance \(d(x,y) = \max\{|x|, |y|\} = 1\), we can analyze the problem step by step.

### Step 1: Understand the Definition of Distance
The distance \(d(x,y)\) is defined as the maximum of the absolute values of \(x\) and \(y\):
\[ d(x,y) = \max\{|x|, |y|\} \]
This means that \(d(x,y)\) will return the larger value between \(|x|\) and \(|y|\).

### Step 2: Set Up the Equation
We are given that \(d(x,y) = 1\). Therefore, we can write:
\[ \max\{|x|, |y|\} = 1 \]

### Step 3: Analyze Cases
We need to consider two cases based on the definition of the maximum function.

#### Case 1: \(|x| \geq |y|\)
In this case, the maximum value is \(|x|\):
\[ |x| = 1 \]
This implies:
\[ x = 1 \quad \text{or} \quad x = -1 \]
For these values of \(x\), \(y\) can take any value such that \(|y| \leq 1\). Thus, the points for this case are:
- For \(x = 1\): \((1, y)\) where \(-1 \leq y \leq 1\)
- For \(x = -1\): \((-1, y)\) where \(-1 \leq y \leq 1\)

#### Case 2: \(|y| > |x|\)
In this case, the maximum value is \(|y|\):
\[ |y| = 1 \]
This implies:
\[ y = 1 \quad \text{or} \quad y = -1 \]
For these values of \(y\), \(x\) can take any value such that \(|x| \leq 1\). Thus, the points for this case are:
- For \(y = 1\): \((x, 1)\) where \(-1 \leq x \leq 1\)
- For \(y = -1\): \((x, -1)\) where \(-1 \leq x \leq 1\)

### Step 4: Combine the Results
From both cases, we can summarize the locus of points:
- Vertical line segments at \(x = 1\) and \(x = -1\) for \(-1 \leq y \leq 1\).
- Horizontal line segments at \(y = 1\) and \(y = -1\) for \(-1 \leq x \leq 1\).

### Step 5: Identify the Shape
Plotting these points on the Cartesian plane, we can see that the locus forms a square with vertices at:
- \((1, 1)\)
- \((1, -1)\)
- \((-1, 1)\)
- \((-1, -1)\)

### Conclusion
The locus of the point \((x,y)\) where \(d(x,y) = 1\) is a square.