If plane `2x+3y+6z+k=0` is tangent to the sphere `x^(2)+y^(2)+z^(2)+2x-2y+2z-6=0`, then a value of k is (a) 26 (b) 16 (c) -26 (d) none of these

Options

A. 26
B. 16
C. `-26`
D. none of these

Explanation

To find the value of \( k \) such that the plane \( 2x + 3y + 6z + k = 0 \) is tangent to the sphere given by the equation \( x^2 + y^2 + z^2 + 2x - 2y + 2z - 6 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation of the sphere in standard form
The equation of the sphere is given as:
\[ x^2 + y^2 + z^2 + 2x - 2y + 2z - 6 = 0 \]
We can complete the square for each variable:

1. For \( x^2 + 2x \):
\[ x^2 + 2x = (x + 1)^2 - 1 \]

2. For \( y^2 - 2y \):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

3. For \( z^2 + 2z \):
\[ z^2 + 2z = (z + 1)^2 - 1 \]

Now substituting these back into the sphere's equation:
\[ ((x + 1)^2 - 1) + ((y - 1)^2 - 1) + ((z + 1)^2 - 1) - 6 = 0 \]
This simplifies to:
\[ (x + 1)^2 + (y - 1)^2 + (z + 1)^2 - 9 = 0 \]
Thus, we can rewrite it as:
\[ (x + 1)^2 + (y - 1)^2 + (z + 1)^2 = 9 \]

### Step 2: Identify the center and radius of the sphere
From the standard form of the sphere's equation \( (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 \):
- The center of the sphere is \( (-1, 1, -1) \)
- The radius \( r = 3 \) (since \( r^2 = 9 \))

### Step 3: Find the distance from the center of the sphere to the plane
The distance \( D \) from a point \( (x_1, y_1, z_1) \) to the plane \( ax + by + cz + d = 0 \) is given by:
\[ D = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]
For our plane \( 2x + 3y + 6z + k = 0 \):
- \( a = 2, b = 3, c = 6, d = k \)
- The center of the sphere \( (x_1, y_1, z_1) = (-1, 1, -1) \)

Substituting these values:
\[ D = \frac{|2(-1) + 3(1) + 6(-1) + k|}{\sqrt{2^2 + 3^2 + 6^2}} \]
Calculating the numerator:
\[ D = \frac{|-2 + 3 - 6 + k|}{\sqrt{4 + 9 + 36}} = \frac{|k - 5|}{7} \]

### Step 4: Set the distance equal to the radius
Since the plane is tangent to the sphere, the distance \( D \) must equal the radius \( r \):
\[ \frac{|k - 5|}{7} = 3 \]
Multiplying both sides by 7:
\[ |k - 5| = 21 \]

### Step 5: Solve for \( k \)
This absolute value equation gives us two cases:
1. \( k - 5 = 21 \)
\[ k = 26 \]
2. \( k - 5 = -21 \)
\[ k = -16 \]

### Conclusion
The possible values for \( k \) are \( 26 \) and \( -16 \). However, since the question asks for a value of \( k \) and provides options, we find that \( k = 26 \) is one of the options.

Thus, the answer is:
\[ \text{(a) } 26 \]