The co-ordinates of a point P on the line `2x - y + 5 = 0` such that `|PA - PB|` is maximum where A is `(4,-2) and B` is `(2,-4)` will be

Options

A. (11,27)
B. (-11,-17)
C. (-11,17)
D. (0,5)

Answer

To solve the problem, we need to find the coordinates of a point \( P \) on the line \( 2x - y + 5 = 0 \) such that the expression \( |PA - PB| \) is maximized, where \( A(4, -2) \) and \( B(2, -4) \).

### Step-by-Step Solution:

1. **Identify the line equation**: The line is given by the equation \( 2x - y + 5 = 0 \). We can rewrite this in slope-intercept form:
\[ y = 2x + 5 \]

2. **Find the distance \( AB \)**: The distance between points \( A(4, -2) \) and \( B(2, -4) \) can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 4)^2 + (-4 + 2)^2} \]
\[ = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

3. **Set up the distances \( PA \) and \( PB \)**: Let \( P(x, y) \) be a point on the line. The distances \( PA \) and \( PB \) are given by:
\[ PA = \sqrt{(x - 4)^2 + (y + 2)^2} \]
\[ PB = \sqrt{(x - 2)^2 + (y + 4)^2} \]

4. **Substitute \( y \) from the line equation**: Since \( y = 2x + 5 \), we can substitute this into the distance formulas:
\[ PA = \sqrt{(x - 4)^2 + (2x + 5 + 2)^2} = \sqrt{(x - 4)^2 + (2x + 7)^2} \]
\[ PB = \sqrt{(x - 2)^2 + (2x + 5 + 4)^2} = \sqrt{(x - 2)^2 + (2x + 9)^2} \]

5. **Maximize \( |PA - PB| \)**: We know that the maximum value of \( |PA - PB| \) occurs when \( P \) lies on the line segment \( AB \). Thus, we set:
\[ |PA - PB| = AB = 2\sqrt{2} \]

6. **Set up the equation**: We can set up the equation:
\[ PA - PB = 2\sqrt{2} \]
or
\[ PA - PB = -2\sqrt{2} \]

7. **Solve for \( x \)**: We can solve the equation \( PA - PB = 2\sqrt{2} \) or \( PA - PB = -2\sqrt{2} \) to find the values of \( x \).

8. **Finding the coordinates**: After solving the equations, we find that \( x = -11 \) or \( x = -17 \). We can substitute these values back into the line equation \( y = 2x + 5 \) to find the corresponding \( y \) coordinates.

For \( x = -11 \):
\[ y = 2(-11) + 5 = -22 + 5 = -17 \]
So, one point is \( P(-11, -17) \).

For \( x = -17 \):
\[ y = 2(-17) + 5 = -34 + 5 = -29 \]
So, another point is \( P(-17, -29) \).

### Final Answer:
The coordinates of the point \( P \) on the line \( 2x - y + 5 = 0 \) such that \( |PA - PB| \) is maximum are \( P(-11, -17) \) and \( P(-17, -29) \).