Find the equation of the sphere whose centre has the position vector `3hat(i)+6hat(j)-4hat(k)` and which touches the plane `r*(2hat(i)-2hat(j)-hat(k))=10`.

Answer

To find the equation of the sphere whose center has the position vector \( \vec{C} = 3\hat{i} + 6\hat{j} - 4\hat{k} \) and which touches the plane given by \( \vec{r} \cdot (2\hat{i} - 2\hat{j} - \hat{k}) = 10 \), we can follow these steps:

### Step 1: Convert the plane equation to Cartesian form
The plane equation in vector form is given as:
\[ \vec{r} \cdot (2\hat{i} - 2\hat{j} - \hat{k}) = 10 \]
This can be expressed in Cartesian coordinates as:
\[ 2x - 2y - z = 10 \]
or rearranging gives:
\[ 2x - 2y - z - 10 = 0 \]

### Step 2: Identify the center of the sphere
The center of the sphere is given by the position vector:
\[ \vec{C} = (3, 6, -4) \]
Thus, the coordinates of the center are:
\[ (x_1, y_1, z_1) = (3, 6, -4) \]

### Step 3: Calculate the distance from the center 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 the formula:
\[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
For our plane \( 2x - 2y - z - 10 = 0 \), we have:
- \( A = 2 \)
- \( B = -2 \)
- \( C = -1 \)
- \( D = -10 \)

Substituting in the coordinates of the center:
\[ d = \frac{|2(3) - 2(6) - (-4) - 10|}{\sqrt{2^2 + (-2)^2 + (-1)^2}} \]
Calculating the numerator:
\[ = |6 - 12 + 4 - 10| = |-12| = 12 \]
Calculating the denominator:
\[ = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]
Thus, the distance \( d \) is:
\[ d = \frac{12}{3} = 4 \]

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

### Step 5: Write the equation of the sphere
The equation of a sphere with center \( (x_1, y_1, z_1) \) and radius \( r \) is given by:
\[ (x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 = r^2 \]
Substituting the values:
\[ (x - 3)^2 + (y - 6)^2 + (z + 4)^2 = 4^2 \]
This simplifies to:
\[ (x - 3)^2 + (y - 6)^2 + (z + 4)^2 = 16 \]

### Final Answer
The equation of the sphere is:
\[ (x - 3)^2 + (y - 6)^2 + (z + 4)^2 = 16 \]