A(2,3) B (-3,4)are two points P moves such that the area of `DeltaPAB` is 8.5 square units, then find the locus of P

Answer

To find the locus of point P such that the area of triangle PAB is 8.5 square units, we can follow these steps:

### Step 1: Set Up the Area Formula
The area of triangle formed by points A(2, 3), B(-3, 4), and P(x, y) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
where \( (x_1, y_1) = (2, 3) \), \( (x_2, y_2) = (-3, 4) \), and \( (x_3, y_3) = (x, y) \).

### Step 2: Substitute the Points into the Area Formula
Substituting the coordinates of points A, B, and P into the area formula gives:
\[ \text{Area} = \frac{1}{2} \left| 2(4 - y) + (-3)(y - 3) + x(3 - 4) \right| \]

### Step 3: Simplify the Expression
Expanding the expression inside the absolute value:
\[ = \frac{1}{2} \left| 8 - 2y - 3y + 9 - x \right| \]
\[ = \frac{1}{2} \left| 17 - 5y - x \right| \]

### Step 4: Set the Area Equal to 8.5
Since we want the area to be 8.5 square units, we set up the equation:
\[ \frac{1}{2} \left| 17 - 5y - x \right| = 8.5 \]

### Step 5: Multiply Both Sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ \left| 17 - 5y - x \right| = 17 \]

### Step 6: Remove the Absolute Value
This gives us two cases to consider:
1. \( 17 - 5y - x = 17 \)
2. \( 17 - 5y - x = -17 \)

### Step 7: Solve Each Case
**Case 1:**
\[ 17 - 5y - x = 17 \implies -5y - x = 0 \implies x + 5y = 0 \implies x = -5y \]

**Case 2:**
\[ 17 - 5y - x = -17 \implies -5y - x = -34 \implies x + 5y = 34 \implies x = 34 - 5y \]

### Step 8: Combine the Results
The two equations we found are:
1. \( x = -5y \)
2. \( x = 34 - 5y \)

These represent two lines in the coordinate plane, which together form the locus of point P.

### Final Result
Thus, the locus of point P is given by the equations:
\[ x + 5y = 0 \quad \text{and} \quad x + 5y = 34 \]