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

Options

A. 268
B. 244
C. 212
D. 252

Answer

To find the square of the length of the segment intercepted 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: Find the intersection points of the lines with the line segment bisecting at (6, 8).

1. **Equation of Line 1**: \(x + 6y - 13 = 0\)
- Rearranging gives: \(x = 13 - 6y\)

2. **Equation of Line 2**: \(x - y + 3 = 0\)
- Rearranging gives: \(x = y - 3\)

### Step 2: Set up the midpoint formula.

The midpoint \(M\) of the segment \(AB\) is given by:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Where \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the intersection points of the lines.

Given that the midpoint is \((6, 8)\), we can set up the following equations:
\[ \frac{x_1 + x_2}{2} = 6 \quad \text{(1)} \]
\[ \frac{y_1 + y_2}{2} = 8 \quad \text{(2)} \]

### Step 3: Solve for \(x_1 + x_2\) and \(y_1 + y_2\).

From equation (1):
\[ x_1 + x_2 = 12 \quad \text{(3)} \]

From equation (2):
\[ y_1 + y_2 = 16 \quad \text{(4)} \]

### Step 4: Substitute \(x_1\) and \(x_2\) in terms of \(y_1\) and \(y_2\).

From the first line:
\[ x_1 = 13 - 6y_1 \quad \text{(5)} \]
From the second line:
\[ x_2 = y_2 - 3 \quad \text{(6)} \]

### Step 5: Substitute equations (5) and (6) into equation (3).

Substituting into equation (3):
\[ (13 - 6y_1) + (y_2 - 3) = 12 \]
Simplifying gives:
\[ 10 - 6y_1 + y_2 = 12 \]
\[ y_2 - 6y_1 = 2 \quad \text{(7)} \]

### Step 6: Solve equations (4) and (7) simultaneously.

From equation (4):
\[ y_2 = 16 - y_1 \quad \text{(8)} \]

Substituting (8) into (7):
\[ (16 - y_1) - 6y_1 = 2 \]
\[ 16 - 7y_1 = 2 \]
\[ 7y_1 = 14 \implies y_1 = 2 \]

### Step 7: Find \(y_2\).

Substituting \(y_1 = 2\) into (8):
\[ y_2 = 16 - 2 = 14 \]

### Step 8: Find \(x_1\) and \(x_2\).

Using (5) to find \(x_1\):
\[ x_1 = 13 - 6(2) = 1 \]

Using (6) to find \(x_2\):
\[ x_2 = 14 - 3 = 11 \]

### Step 9: Calculate the length of segment \(AB\).

The coordinates of points \(A\) and \(B\) are \(A(1, 2)\) and \(B(11, 14)\). The length \(AB\) is given by:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(11 - 1)^2 + (14 - 2)^2} \]
Calculating gives:
\[ AB = \sqrt{10^2 + 12^2} = \sqrt{100 + 144} = \sqrt{244} \]

### Step 10: Find the square of the length of the segment.

The square of the length of segment \(AB\) is:
\[ AB^2 = 244 \]

### Final Answer:
The square of the length of the segment is \(244\).