A lion moves in the region given by the graph `y -|y|- x+ |x|=0`. curve a person can move so that he does not encounter lion -

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 of determining the region where a person can move without encountering a lion, we start with the equation given:

\[ y - |y| - x + |x| = 0 \]

We will analyze this equation by considering the different cases for the signs of \(x\) and \(y\).

### Step 1: Analyze the equation based on the signs of \(x\) and \(y\)

1. **Case 1:** \(x \geq 0\) and \(y \geq 0\)
- Here, \(|y| = y\) and \(|x| = x\).
- The equation simplifies to:
\[ y - y - x + x = 0 \implies 0 = 0 \]
- This case provides no restrictions.

2. **Case 2:** \(x < 0\) and \(y < 0\)
- Here, \(|y| = -y\) and \(|x| = -x\).
- The equation simplifies to:
\[ y + y - x - x = 0 \implies 2y - 2x = 0 \implies y = x \]
- This means that in the third quadrant, the lion moves along the line \(y = x\).

3. **Case 3:** \(x \geq 0\) and \(y < 0\)
- Here, \(|y| = -y\) and \(|x| = x\).
- The equation simplifies to:
\[ y + y - x + x = 0 \implies 2y = 0 \implies y = 0 \]
- This means that when \(x \geq 0\), the lion is on the x-axis.

4. **Case 4:** \(x < 0\) and \(y \geq 0\)
- Here, \(|y| = y\) and \(|x| = -x\).
- The equation simplifies to:
\[ y - y - x - x = 0 \implies -2x = 0 \implies x = 0 \]
- This means that when \(y \geq 0\), the lion is on the y-axis.

### Step 2: Summarize the lion's movement region

From the analysis, we can summarize the lion's movement:
- In the third quadrant, the lion moves along the line \(y = x\).
- On the x-axis for \(x \geq 0\).
- On the y-axis for \(y \geq 0\).

### Step 3: Determine the safe regions for the person

To avoid encountering the lion, the person should stay away from:
- The line \(y = x\) in the third quadrant.
- The positive x-axis (where \(y = 0\) for \(x \geq 0\)).
- The positive y-axis (where \(x = 0\) for \(y \geq 0\)).

### Step 4: Identify a safe path

A person can move in the regions:
- Above the x-axis (for \(x \geq 0\) and \(y > 0\)).
- Below the line \(y = x\) in the third quadrant (for \(x < 0\) and \(y < 0\)).
- Below the y-axis (for \(x < 0\) and \(y > 0\)).

### Conclusion

Thus, the safest path for a person is to avoid the axes and the line \(y = x\) in the third quadrant.