The coordinates of three points O, A, B are (0, 0), (0,4) and (6, 0) respectively. A point P moves so that the area of `Delta POA` is always twice the area of `Delta POB`. Find the equation to both parts of the locus of P.

Answer

To find the locus of the point P such that the area of triangle POA is always twice the area of triangle POB, we can follow these steps:

### Step 1: Identify the points and their coordinates
The points O, A, and B have the following coordinates:
- O (0, 0)
- A (0, 4)
- B (6, 0)

Let P be a point with coordinates (x, y).

### Step 2: Calculate the area of triangle POA
The area of triangle POA can be calculated using the formula for the area of a triangle formed by three points:
\[ \text{Area}_{POA} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base OA is vertical (length 4) and the height from P to OA is the x-coordinate of P (x). Thus,
\[ \text{Area}_{POA} = \frac{1}{2} \times 4 \times x = 2x \]

### Step 3: Calculate the area of triangle POB
Similarly, the area of triangle POB can be calculated. The base OB is horizontal (length 6) and the height from P to OB is the y-coordinate of P (y). Thus,
\[ \text{Area}_{POB} = \frac{1}{2} \times 6 \times y = 3y \]

### Step 4: Set up the equation based on the given condition
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:
\[ 2x = 2 \times 3y \]
This simplifies to:
\[ 2x = 6y \]

### Step 5: Simplify the equation
Dividing both sides by 2 gives:
\[ x = 3y \]

### Step 6: Write the equation of the locus
The equation \(x = 3y\) can be rearranged to:
\[ x - 3y = 0 \]

### Step 7: Consider both parts of the locus
Since the area can also be negative (considering the orientation of the triangle), we can also have:
\[ x = -3y \]
This leads to the second part of the locus:
\[ x + 3y = 0 \]

### Final Equation of the Locus
Combining both parts, the locus of point P is given by:
\[ (x - 3y)(x + 3y) = 0 \]