Let `0-=(0,0),A-=(0,4),B-=(6,0)dot` Let `P` be a moving point such that the area of triangle `P O A` is two times the area of triangle `P O B` . The locus of `P` will be a straight line whose equation can be `x+3y=0` (b) `x+2y=0` `2x-3y=0` (d) `3y-x=0`

Options

A. `x+3y=0`
B. `x+2y=0`
C. `2x-3y=0`
D. `3y-x=0`

Explanation

To solve the problem, we need to find the locus of the point \( P \) such that the area of triangle \( POA \) is twice the area of triangle \( POB \).

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( O = (0, 0) \)
- Let \( A = (0, 4) \)
- Let \( B = (6, 0) \)
- Let \( P = (x, y) \) be the moving point.

2. **Calculate the Area of Triangle \( POA \)**:
- The area of triangle \( POA \) can be calculated using the formula:
\[ \text{Area}_{POA} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base \( OA \) is the vertical distance from \( O \) to \( A \), which is \( 4 \). The height is the x-coordinate of point \( P \), which is \( x \).
- Therefore, the area of triangle \( POA \) is:
\[ \text{Area}_{POA} = \frac{1}{2} \times 4 \times x = 2x \]

3. **Calculate the Area of Triangle \( POB \)**:
- Similarly, the area of triangle \( POB \) is:
\[ \text{Area}_{POB} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base \( OB \) is the horizontal distance from \( O \) to \( B \), which is \( 6 \). The height is the y-coordinate of point \( P \), which is \( y \).
- Therefore, the area of triangle \( POB \) is:
\[ \text{Area}_{POB} = \frac{1}{2} \times 6 \times y = 3y \]

4. **Set up the Equation**:
- According to the problem, the area of triangle \( POA \) is twice the area of triangle \( POB \):
\[ \text{Area}_{POA} = 2 \times \text{Area}_{POB} \]
- Substituting the areas we calculated:
\[ 2x = 2 \times 3y \]
- Simplifying this gives:
\[ 2x = 6y \]
- Dividing both sides by 2:
\[ x = 3y \]

5. **Rearranging the Equation**:
- Rearranging the equation \( x - 3y = 0 \) gives us the locus of point \( P \).

### Conclusion:
The locus of point \( P \) is given by the equation:
\[ x - 3y = 0 \]
Thus, the correct option is (d) \( 3y - x = 0 \).