If the segment between thelines `x+6y-13=0` and `x-y+3=0` is bisected at `(6, 8)` then length of segment is :

Options

A. `sqrt(268)`
B. `sqrt(244)`
C. `sqrt(212)`
D. `sqrt(252)`

Explanation

To find the length of the segment between the lines \(x + 6y - 13 = 0\) and \(x - y + 3 = 0\) that is bisected at the point \((6, 8)\), we can follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are:
1. Line 1: \(L_1: x + 6y - 13 = 0\)
2. Line 2: \(L_2: x - y + 3 = 0\)

### Step 2: Express the points of intersection
Let the points where the segment intersects line \(L_1\) and line \(L_2\) be \(A(x_1, y_1)\) and \(B(x_2, y_2)\) respectively.

From line \(L_1\):
\[ x_1 + 6y_1 - 13 = 0 \implies x_1 = 13 - 6y_1 \]
Thus, point \(A\) can be expressed as:
\[ A(13 - 6y_1, y_1) \]

From line \(L_2\):
\[ x_2 - y_2 + 3 = 0 \implies x_2 = y_2 - 3 \]
Thus, point \(B\) can be expressed as:
\[ B(y_2 - 3, y_2) \]

### Step 3: Use the midpoint formula
The midpoint \(M\) of segment \(AB\) is given by:
\[ M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = (6, 8) \]

### Step 4: Set up equations from the midpoint
From the midpoint coordinates, we have two equations:
1. \(\frac{(13 - 6y_1) + (y_2 - 3)}{2} = 6\)
2. \(\frac{y_1 + y_2}{2} = 8\)

### Step 5: Solve the first equation
Multiplying the first equation by 2:
\[ (13 - 6y_1) + (y_2 - 3) = 12 \]
This simplifies to:
\[ 10 - 6y_1 + y_2 = 12 \implies y_2 - 6y_1 = 2 \tag{1} \]

### Step 6: Solve the second equation
Multiplying the second equation by 2:
\[ y_1 + y_2 = 16 \tag{2} \]

### Step 7: Solve the system of equations
Now we can solve equations (1) and (2):
From (2), we can express \(y_2\) in terms of \(y_1\):
\[ y_2 = 16 - y_1 \]
Substituting into (1):
\[ (16 - y_1) - 6y_1 = 2 \implies 16 - 7y_1 = 2 \implies 7y_1 = 14 \implies y_1 = 2 \]
Now substituting \(y_1 = 2\) back into (2):
\[ 2 + y_2 = 16 \implies y_2 = 14 \]

### Step 8: Find \(x_1\) and \(x_2\)
Now we can find \(x_1\) and \(x_2\):
\[ x_1 = 13 - 6(2) = 1 \implies A(1, 2) \]
\[ x_2 = 14 - 3 = 11 \implies B(11, 14) \]

### Step 9: Calculate the length of segment \(AB\)
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(11 - 1)^2 + (14 - 2)^2} \]
Calculating:
\[ = \sqrt{10^2 + 12^2} = \sqrt{100 + 144} = \sqrt{244} = \sqrt{4 \times 61} = 2\sqrt{61} \]

### Final Answer
The length of the segment is \(2\sqrt{61}\).