Find the equation of sphere if it touches the plane `rcdot(2hat(i)-2hat(j)-hat(k))=0` and the position vector of its centre is `3hat(i)+6hat(j)-hat(k).`

Answer

To find the equation of the sphere that touches the given plane and has a specified center, we will follow these steps:

### Step 1: Identify the center of the sphere and the normal vector of the plane
The position vector of the center of the sphere is given as:
\[ \vec{C} = 3\hat{i} + 6\hat{j} - \hat{k} \]
The equation of the plane is given as:
\[ \vec{r} \cdot (2\hat{i} - 2\hat{j} - \hat{k}) = 0 \]
From this equation, we can identify the normal vector of the plane:
\[ \vec{n} = 2\hat{i} - 2\hat{j} - \hat{k} \]

### Step 2: Calculate the distance from the center of the sphere to the plane
The distance \(d\) from a point (the center of the sphere) to a plane can be calculated using the formula:
\[ d = \frac{|\vec{n} \cdot \vec{C}|}{|\vec{n}|} \]
First, we need to calculate \(\vec{n} \cdot \vec{C}\):
\[ \vec{n} \cdot \vec{C} = (2\hat{i} - 2\hat{j} - \hat{k}) \cdot (3\hat{i} + 6\hat{j} - \hat{k}) = 2 \cdot 3 + (-2) \cdot 6 + (-1) \cdot (-1) \]
Calculating this gives:
\[ = 6 - 12 + 1 = -5 \]
Now, we calculate the magnitude of \(\vec{n}\):
\[ |\vec{n}| = \sqrt{2^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]
Now, substituting these values into the distance formula:
\[ d = \frac{|-5|}{3} = \frac{5}{3} \]

### Step 3: Determine the radius of the sphere
Since the sphere touches the plane, the distance \(d\) from the center of the sphere to the plane is equal to the radius \(r\) of the sphere:
\[ r = \frac{5}{3} \]

### Step 4: Write the equation of the sphere
The general 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 \(x_1 = 3\), \(y_1 = 6\), \(z_1 = -1\), and \(r = \frac{5}{3}\):
\[ (x - 3)^2 + (y - 6)^2 + (z + 1)^2 = \left(\frac{5}{3}\right)^2 \]
Calculating \(r^2\):
\[ \left(\frac{5}{3}\right)^2 = \frac{25}{9} \]
Thus, the equation of the sphere becomes:
\[ (x - 3)^2 + (y - 6)^2 + (z + 1)^2 = \frac{25}{9} \]

### Final Answer
The equation of the sphere is:
\[ (x - 3)^2 + (y - 6)^2 + (z + 1)^2 = \frac{25}{9} \]