The centre of the circle given by `vec r* (hat i+ 2hat j + 2hat k) = 15 and |vec r-(hat j + 2hat k)| = 4` is

Answer

To find the center of the circle defined by the given equations, we will follow these steps:

### Step 1: Convert the equations to Cartesian form

The first equation is given as:
\[ \vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) = 15 \]
This can be expanded in Cartesian coordinates as:
\[ x + 2y + 2z = 15 \]
This represents a plane.

The second equation is given as:
\[ |\vec{r} - (\hat{j} + 2\hat{k})| = 4 \]
This can be rewritten as:
\[ \sqrt{(x - 0)^2 + (y - 1)^2 + (z - 2)^2} = 4 \]
Squaring both sides gives us:
\[ (x - 0)^2 + (y - 1)^2 + (z - 2)^2 = 16 \]
This represents a sphere with center at (0, 1, 2) and radius 4.

### Step 2: Identify the center of the sphere

From the equation of the sphere, we can identify its center:
\[ \text{Center of sphere} = (0, 1, 2) \]

### Step 3: Find the normal vector to the plane

The normal vector to the plane \(x + 2y + 2z = 15\) can be derived from the coefficients of \(x\), \(y\), and \(z\):
\[ \text{Normal vector} = (1, 2, 2) \]

### Step 4: Set up the line joining the center of the sphere to the plane

The line joining the center of the sphere (0, 1, 2) to any point on the plane can be represented parametrically as:
\[ \begin{align*} x &= 0 + t \cdot 1 = t \\ y &= 1 + t \cdot 2 = 1 + 2t \\ z &= 2 + t \cdot 2 = 2 + 2t \end{align*} \]

### Step 5: Substitute into the plane equation

Substituting the parametric equations into the plane equation:
\[ t + 2(1 + 2t) + 2(2 + 2t) = 15 \]
Expanding this gives:
\[ t + 2 + 4t + 4 + 4t = 15 \]
Combining like terms:
\[ 9t + 6 = 15 \]
Solving for \(t\):
\[ 9t = 9 \implies t = 1 \]

### Step 6: Find the coordinates of the center of the circle

Now substituting \(t = 1\) back into the parametric equations:
\[ \begin{align*} x &= 1 \\ y &= 1 + 2(1) = 3 \\ z &= 2 + 2(1) = 4 \end{align*} \]
Thus, the center of the circle is:
\[ \text{Center of circle} = (1, 3, 4) \]

### Final Answer:
The center of the circle is \((1, 3, 4)\).

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