What is the equation of the sphere which has its centre at (6,-1,2) and touches the plane `2x-y+2z-2=0`?

Options

A. `x^2+y^2+z^2+12x-2y+4z+16=0`
B. `x^2+y^2+z^2+12x-2y+4z-16=0`
C. `x^2+y^2+z^2-12x+2y-4z+16=0`
D. `x^2+y^2+z^2-12x+2y-4z+25=0`

Explanation

To find the equation of the sphere with its center at (6, -1, 2) that touches the plane given by the equation \(2x - y + 2z - 2 = 0\), we will follow these steps:

### Step 1: Identify the center of the sphere and the equation of the plane
The center of the sphere is given as \(C(6, -1, 2)\) and the equation of the plane is \(2x - y + 2z - 2 = 0\).

### Step 2: Use the formula for the distance from a point to a plane
The distance \(d\) from a point \((x_1, y_1, z_1)\) to a plane given by the equation \(ax + by + cz + d = 0\) is calculated using the formula:
\[ d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]
In our case, the coefficients from the plane equation \(2x - y + 2z - 2 = 0\) are:
- \(a = 2\)
- \(b = -1\)
- \(c = 2\)
- \(d = -2\)

### Step 3: Substitute the center coordinates into the distance formula
Now, we substitute the center coordinates \(C(6, -1, 2)\) into the distance formula:
\[ d = \frac{|2(6) + (-1)(-1) + 2(2) - 2|}{\sqrt{2^2 + (-1)^2 + 2^2}} \]
Calculating the numerator:
\[ = |12 + 1 + 4 - 2| = |15| = 15 \]
Calculating the denominator:
\[ = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]
Thus, the distance \(d\) is:
\[ d = \frac{15}{3} = 5 \]

### Step 4: Determine the radius of the sphere
Since the sphere touches the plane, the distance \(d\) calculated above is equal to the radius \(r\) of the sphere:
\[ r = 5 \]

### Step 5: Write the equation of the sphere
The standard equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]
Substituting \(h = 6\), \(k = -1\), \(l = 2\), and \(r = 5\):
\[ (x - 6)^2 + (y + 1)^2 + (z - 2)^2 = 5^2 \]
This simplifies to:
\[ (x - 6)^2 + (y + 1)^2 + (z - 2)^2 = 25 \]

### Step 6: Expand the equation
Expanding the equation:
\[ (x^2 - 12x + 36) + (y^2 + 2y + 1) + (z^2 - 4z + 4) = 25 \]
Combining like terms:
\[ x^2 + y^2 + z^2 - 12x + 2y - 4z + 41 = 25 \]
Rearranging gives:
\[ x^2 + y^2 + z^2 - 12x + 2y - 4z + 16 = 0 \]

### Final Answer
The equation of the sphere is:
\[ x^2 + y^2 + z^2 - 12x + 2y - 4z + 16 = 0 \]