The distance of the line x + y - 8 = 0 from (4,1) measured along the direction whose slope is -2 , is

Options

A. `3sqrt5`
B. `6 sqrt5`
C. `2 sqrt5`
D. none of these

Answer

To find the distance of the line \(x + y - 8 = 0\) from the point \((4, 1)\) measured along the direction whose slope is \(-2\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given line and point**:
The line is given by the equation:
\[ x + y - 8 = 0 \]
The point from which we want to measure the distance is \((4, 1)\).

2. **Check if the point lies on the line**:
Substitute the coordinates of the point into the line equation:
\[ 4 + 1 - 8 = -3 \quad (\text{not on the line}) \]
Since the result is not zero, the point \((4, 1)\) does not lie on the line.

3. **Determine the slope of the line**:
The slope of the line \(x + y - 8 = 0\) can be rearranged to the slope-intercept form:
\[ y = -x + 8 \]
Thus, the slope of the line is \(-1\).

4. **Determine the slope of the direction**:
The slope of the direction along which we are measuring the distance is given as \(-2\).

5. **Find the equation of the line in the direction of slope -2 through the point (4, 1)**:
Using point-slope form:
\[ y - 1 = -2(x - 4) \]
Simplifying this:
\[ y - 1 = -2x + 8 \implies 2x + y - 9 = 0 \]

6. **Find the intersection of the two lines**:
We need to solve the system of equations:
\[ x + y - 8 = 0 \quad \text{(1)} \]
\[ 2x + y - 9 = 0 \quad \text{(2)} \]
Subtract equation (1) from equation (2):
\[ (2x + y - 9) - (x + y - 8) = 0 \]
This simplifies to:
\[ 2x + y - 9 - x - y + 8 = 0 \implies x - 1 = 0 \implies x = 1 \]
Substitute \(x = 1\) back into equation (1):
\[ 1 + y - 8 = 0 \implies y = 7 \]
Thus, the point of intersection is \((1, 7)\).

7. **Calculate the distance between the points (4, 1) and (1, 7)**:
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1) = (4, 1)\) and \((x_2, y_2) = (1, 7)\):
\[ d = \sqrt{(1 - 4)^2 + (7 - 1)^2} = \sqrt{(-3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \]

### Final Answer:
The distance from the line \(x + y - 8 = 0\) to the point \((4, 1)\) measured along the direction whose slope is \(-2\) is:
\[ \boxed{3\sqrt{5}} \]