Two roads are represented by the equations `y - x = 6 and x+ y = 8 ` An inspection bungalow has to be so constructed that it is at a distance of 100 from each of the roads . Possible location of the bungalow is given by

Options

A. `(100sqrt(2)+1,7)`
B. `(1-100sqrt(2),7)`
C. `(1,7+100sqrt(2))`
D. `(1,7-100sqrt(2))`

Answer

To find the possible locations of the inspection bungalow that is at a distance of 100 meters from each of the roads represented by the equations \(y - x = 6\) and \(x + y = 8\), we will use the formula for the perpendicular distance from a point to a line.

### Step-by-Step Solution:

1. **Identify the equations of the roads:**
- The first road is given by the equation \(y - x = 6\).
- The second road is given by the equation \(x + y = 8\).

2. **Rearranging the equations:**
- The first equation can be rearranged to the standard form: \(x - y + 6 = 0\).
- The second equation can be rearranged to: \(x + y - 8 = 0\).

3. **Using the distance formula:**
The distance \(d\) from a point \((h, k)\) to a line given by \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]

4. **Calculate the distance from the first road:**
For the first road \(x - y + 6 = 0\) (where \(A = 1, B = -1, C = 6\)):
\[ \text{Distance} = \frac{|1 \cdot h - 1 \cdot k + 6|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k + 6|}{\sqrt{2}} \]
Setting this equal to 100:
\[ \frac{|h - k + 6|}{\sqrt{2}} = 100 \]
Thus, we have:
\[ |h - k + 6| = 100\sqrt{2} \]
This gives us two equations:
\[ h - k + 6 = 100\sqrt{2} \quad \text{(1)} \]
\[ h - k + 6 = -100\sqrt{2} \quad \text{(2)} \]

5. **Calculate the distance from the second road:**
For the second road \(x + y - 8 = 0\) (where \(A = 1, B = 1, C = -8\)):
\[ \text{Distance} = \frac{|1 \cdot h + 1 \cdot k - 8|}{\sqrt{1^2 + 1^2}} = \frac{|h + k - 8|}{\sqrt{2}} \]
Setting this equal to 100:
\[ \frac{|h + k - 8|}{\sqrt{2}} = 100 \]
Thus, we have:
\[ |h + k - 8| = 100\sqrt{2} \]
This gives us two equations:
\[ h + k - 8 = 100\sqrt{2} \quad \text{(3)} \]
\[ h + k - 8 = -100\sqrt{2} \quad \text{(4)} \]

6. **Solving the equations:**
We now have four equations to solve:
- From (1) and (3):
\[ h - k + 6 = 100\sqrt{2} \quad \text{and} \quad h + k - 8 = 100\sqrt{2} \]
Adding these:
\[ 2h - 2 = 200\sqrt{2} \implies h = 100\sqrt{2} + 1 \]
Substituting \(h\) back into (1) gives:
\[ 100\sqrt{2} + 1 - k + 6 = 100\sqrt{2} \implies k = 7 \]
Thus, one possible location is \((100\sqrt{2} + 1, 7)\).

- From (1) and (4):
\[ h - k + 6 = 100\sqrt{2} \quad \text{and} \quad h + k - 8 = -100\sqrt{2} \]
Adding these:
\[ 2h - 2 = -100\sqrt{2} \implies h = -50\sqrt{2} + 1 \]
Substituting \(h\) back into (1) gives:
\[ -50\sqrt{2} + 1 - k + 6 = 100\sqrt{2} \implies k = 7 + 50\sqrt{2} \]
Thus, another possible location is \((-50\sqrt{2} + 1, 7 + 50\sqrt{2})\).

- Similarly, we can find other combinations from (2) and (3) and (2) and (4) to find:
- \((1, 7 + 100\sqrt{2})\)
- \((1, 7 - 100\sqrt{2})\)

7. **Final possible locations:**
The possible locations for the bungalow are:
- \((1, 7 + 100\sqrt{2})\)
- \((1, 7 - 100\sqrt{2})\)
- \((100\sqrt{2} + 1, 7)\)
- \((-100\sqrt{2} + 1, 7)\)