The point on the line `(x-2)/1=(y+3)/(-2)=(z+5)/(-2)` at a distance of 6 from the point `(2,-3,-5)` is (a). `(3,-5,-3)` (b). `(4,-7,-9)` (c). `0,2,-1` (d). none of these

Options

A. (3,-5,-3)
B. (4,-7,-9)
C. (0,2,-1)
D. (-3,5,3)

Explanation

To find the point on the line \((x-2)/1=(y+3)/(-2)=(z+5)/(-2)\) that is at a distance of 6 from the point \((2,-3,-5)\), we can follow these steps:

### Step 1: Parametrize the line
The given equation of the line can be expressed in parametric form. Let \( k \) be the parameter. Then we can write:
\[ x = k + 2, \quad y = -2k - 3, \quad z = -2k - 5 \]

### Step 2: Use the distance formula
The distance \( D \) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is given by:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
We need to find the distance from the point \((2, -3, -5)\) to the point on the line \((k + 2, -2k - 3, -2k - 5)\). Setting the distance equal to 6, we have:
\[ 6 = \sqrt{(k + 2 - 2)^2 + (-2k - 3 + 3)^2 + (-2k - 5 + 5)^2} \]

### Step 3: Simplify the distance equation
This simplifies to:
\[ 6 = \sqrt{(k)^2 + (-2k)^2 + (-2k)^2} \]
\[ 6 = \sqrt{k^2 + 4k^2 + 4k^2} \]
\[ 6 = \sqrt{9k^2} \]
\[ 6 = 3|k| \]

### Step 4: Solve for \( k \)
From \( 6 = 3|k| \), we can solve for \( k \):
\[ |k| = 2 \implies k = 2 \text{ or } k = -2 \]

### Step 5: Find the points corresponding to \( k \)
Now we will find the points on the line for both values of \( k \).

1. For \( k = 2 \):
\[ x = 2 + 2 = 4, \quad y = -2(2) - 3 = -7, \quad z = -2(2) - 5 = -9 \]
So, the point is \( (4, -7, -9) \).

2. For \( k = -2 \):
\[ x = -2 + 2 = 0, \quad y = -2(-2) - 3 = 1, \quad z = -2(-2) - 5 = -1 \]
So, the point is \( (0, 1, -1) \).

### Conclusion
The points on the line that are at a distance of 6 from the point \((2, -3, -5)\) are \( (4, -7, -9) \) and \( (0, 1, -1) \). Among the given options, the point \( (4, -7, -9) \) is correct.

### Final Answer
(b) \( (4, -7, -9) \)