The distances of the point (1,-5,9) from the plane `x-y+z=5` measured along a straight line x=y=z is `2sqrt(3)` k , then the value of k is

Options

A. 5
B. 6
C. `sqrt(3)`
D. 4

Answer

To solve the problem, we need to find the value of \( k \) such that the distance from the point \( (1, -5, 9) \) to the plane \( x - y + z = 5 \) measured along the line \( x = y = z \) is \( 2\sqrt{3}k \).

### Step-by-Step Solution:

1. **Identify the direction ratios of the line**:
The line \( x = y = z \) can be represented as having direction ratios \( (1, 1, 1) \).

2. **Parametrize the line**:
We can express points on the line in terms of a parameter \( \lambda \):
\[ x = 1 + \lambda, \quad y = -5 + \lambda, \quad z = 9 + \lambda \]

3. **Substitute into the plane equation**:
We need to find the point on the line that lies on the plane \( x - y + z = 5 \):
\[ (1 + \lambda) - (-5 + \lambda) + (9 + \lambda) = 5 \]
Simplifying this gives:
\[ 1 + \lambda + 5 - \lambda + 9 + \lambda = 5 \\ 15 = 5 \]
This indicates that we need to solve for \( \lambda \):
\[ 1 + 5 + 9 + \lambda - \lambda = 5 \\ 15 - \lambda = 5 \\ \lambda = 10 \]

4. **Find the coordinates of the point on the line**:
Substitute \( \lambda = -10 \) back into the parameterization:
\[ x = 1 - 10 = -9, \quad y = -5 - 10 = -15, \quad z = 9 - 10 = -1 \]
So the point on the line is \( (-9, -15, -1) \).

5. **Calculate the distance between the points**:
The distance \( d \) between the point \( (1, -5, 9) \) and the point \( (-9, -15, -1) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{((-9) - 1)^2 + ((-15) - (-5))^2 + ((-1) - 9)^2} \]
This simplifies to:
\[ d = \sqrt{(-10)^2 + (-10)^2 + (-10)^2} = \sqrt{100 + 100 + 100} = \sqrt{300} = 10\sqrt{3} \]

6. **Set the distance equal to the given expression**:
We know from the problem statement that this distance is also given as \( 2\sqrt{3}k \):
\[ 10\sqrt{3} = 2\sqrt{3}k \]

7. **Solve for \( k \)**:
Dividing both sides by \( 2\sqrt{3} \):
\[ k = \frac{10\sqrt{3}}{2\sqrt{3}} = 5 \]

### Final Answer:
The value of \( k \) is \( 5 \).