Consider the points `A(0,1)a n dB(2,0),a n dP` be a point on the line `4x+3y+9=0` . The coordinates of `P` such that `|P A-P B|` is maximum are (a) `(-(24)/5,(17)/5)` (b) `(-(84)/5,(13)/5)` (c) `((31)/7,(31)/7)` (d) `(-3,0)`

Explanation

To solve the problem, we need to find the coordinates of point \( P \) such that the expression \( |PA - PB| \) is maximized, where \( A(0, 1) \) and \( B(2, 0) \) are fixed points, and \( P \) lies on the line given by the equation \( 4x + 3y + 9 = 0 \).

### Step 1: Find the equation of line AB
The first step is to find the equation of the line passing through points \( A(0, 1) \) and \( B(2, 0) \).

The slope \( m \) of line \( AB \) can be calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{2 - 0} = -\frac{1}{2} \]

Using point-slope form of the line equation, we can write:
\[ y - y_1 = m(x - x_1) \implies y - 1 = -\frac{1}{2}(x - 0) \implies y = -\frac{1}{2}x + 1 \]

To express this in standard form, we can multiply through by 2:
\[ 2y + x - 2 = 0 \implies x + 2y - 2 = 0 \]

### Step 2: Set up the equations
We have two equations now:
1. The line \( AB: x + 2y - 2 = 0 \) (Equation 1)
2. The line \( P: 4x + 3y + 9 = 0 \) (Equation 2)

### Step 3: Solve the system of equations
To find the intersection point \( P \), we need to solve these two equations simultaneously.

From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 2 - 2y \]

Substituting this into Equation 2:
\[ 4(2 - 2y) + 3y + 9 = 0 \]
Expanding this:
\[ 8 - 8y + 3y + 9 = 0 \implies 17 - 5y = 0 \implies 5y = 17 \implies y = \frac{17}{5} \]

Now substituting \( y \) back into the expression for \( x \):
\[ x = 2 - 2\left(\frac{17}{5}\right) = 2 - \frac{34}{5} = \frac{10}{5} - \frac{34}{5} = -\frac{24}{5} \]

### Step 4: Conclusion
Thus, the coordinates of point \( P \) are:
\[ P\left(-\frac{24}{5}, \frac{17}{5}\right) \]

### Step 5: Verify the options
Now we check the options provided:
(a) \((-24/5, 17/5)\) - This matches our result.

Thus, the correct answer is option (a).