If high voltage current is applied on the field given by the graph `y+|y|-x-|x|=0.` on which of the following curve a person can move so that the remains safe ?

Options

A. `y=x ^(2)`
B. `y=sgn(-e ^(2))`
C. `y= log _(1//3) x`
D. `y=m +|x| ,mgt 3`

Answer

To solve the problem given by the equation \( y + |y| - x - |x| = 0 \), we will analyze it in different quadrants based on the signs of \( x \) and \( y \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation can be rewritten as:
\[ y + |y| = x + |x| \]
This means we need to consider different cases based on the signs of \( x \) and \( y \).

2. **Case 1: \( x \geq 0 \) and \( y \geq 0 \)**:
Here, \( |y| = y \) and \( |x| = x \):
\[ y + y = x + x \implies 2y = 2x \implies y = x \]
This represents the line \( y = x \) in the first quadrant.

3. **Case 2: \( x < 0 \) and \( y < 0 \)**:
Here, \( |y| = -y \) and \( |x| = -x \):
\[ y - y = -x - (-x) \implies 0 = 0 \]
This case gives no new information.

4. **Case 3: \( x \geq 0 \) and \( y < 0 \)**:
Here, \( |y| = -y \) and \( |x| = x \):
\[ y - y = x - x \implies 0 = 0 \]
Again, this case gives no new information.

5. **Case 4: \( x < 0 \) and \( y \geq 0 \)**:
Here, \( |y| = y \) and \( |x| = -x \):
\[ y + y = -x + x \implies 2y = 0 \implies y = 0 \]
This represents the line \( y = 0 \) in the second quadrant.

6. **Summary of Results**:
- In the first quadrant: \( y = x \)
- In the second quadrant: \( y = 0 \)
- In the third quadrant: No new information.
- In the fourth quadrant: No new information.

7. **Graphing the Results**:
- The line \( y = x \) intersects the x-axis and y-axis at the origin.
- The line \( y = 0 \) is the x-axis.

8. **Identifying Safe Regions**:
The regions where a person can move safely are outside the lines \( y = x \) and \( y = 0 \).

9. **Analyzing Options**:
- **Option 1**: \( y = x^2 \) - This parabola intersects \( y = x \) and is unsafe.
- **Option 2**: \( y = -1 \) - This line is below \( y = 0 \) and is unsafe.
- **Option 3**: \( y = m + |x| \) where \( m > 3 \) - This line is above \( y = 0 \) and does not intersect \( y = x \) or \( y = 0 \), making it safe.

### Conclusion:
The correct answer is \( y = m + |x| \) where \( m > 3 \).