P is a point lying on line y = x then maximum value of `|PA - PB|`, (where`A-=(1, 3), B-=(5, 2)` ) is

Options

A. (a)`sqrt5`
B. (b)`2sqrt2`
C. (c)`sqrt(17)`
D. (d)`(3)/(sqrt2)`

Explanation

To find the maximum value of \(|PA - PB|\) where \(P\) is a point on the line \(y = x\), and \(A = (1, 3)\) and \(B = (5, 2)\), we can follow these steps:

### Step 1: Understand the problem
We need to find the maximum value of the expression \(|PA - PB|\). Here, \(PA\) is the distance from point \(P\) to point \(A\), and \(PB\) is the distance from point \(P\) to point \(B\).

### Step 2: Set the coordinates of point \(P\)
Since \(P\) lies on the line \(y = x\), we can represent \(P\) as \(P(x, x)\).

### Step 3: Calculate distances \(PA\) and \(PB\)
Using the distance formula, we can calculate the distances \(PA\) and \(PB\):

- The distance \(PA\) from \(P(x, x)\) to \(A(1, 3)\):
\[ PA = \sqrt{(x - 1)^2 + (x - 3)^2} \]

- The distance \(PB\) from \(P(x, x)\) to \(B(5, 2)\):
\[ PB = \sqrt{(x - 5)^2 + (x - 2)^2} \]

### Step 4: Simplify the expressions for \(PA\) and \(PB\)
Calculating \(PA\):
\[ PA = \sqrt{(x - 1)^2 + (x - 3)^2} = \sqrt{(x - 1)^2 + (x^2 - 6x + 9)} = \sqrt{2x^2 - 8x + 10} \]

Calculating \(PB\):
\[ PB = \sqrt{(x - 5)^2 + (x - 2)^2} = \sqrt{(x^2 - 10x + 25) + (x^2 - 4x + 4)} = \sqrt{2x^2 - 14x + 29} \]

### Step 5: Set up the expression for \(|PA - PB|\)
Now we want to maximize \(|PA - PB|\):
\[ |PA - PB| = \left| \sqrt{2x^2 - 8x + 10} - \sqrt{2x^2 - 14x + 29} \right| \]

### Step 6: Use the triangle inequality
From the triangle inequality, we know that:
\[ |PA - PB| \leq AB \]
where \(AB\) is the distance between points \(A\) and \(B\).

### Step 7: Calculate the distance \(AB\)
Using the distance formula:
\[ AB = \sqrt{(5 - 1)^2 + (2 - 3)^2} = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

### Step 8: Conclusion
Thus, the maximum value of \(|PA - PB|\) is equal to the distance \(AB\):
\[ \text{Maximum value of } |PA - PB| = \sqrt{17} \]

### Final Answer
The maximum value of \(|PA - PB|\) is \(\sqrt{17}\).

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