The plane passing through the point `(-2,-2,2)` and contanining the line joining the points `(1,1,1)` and `(1,-1,2)` marks intercepts a,b,c on the axes of coordinates. The value of `a+b+c` is

Options

A. 12
B. 6
C. -4
D. -3

Answer

To solve the problem, we need to find the equation of the plane that passes through the point \((-2, -2, 2)\) and contains the line joining the points \((1, 1, 1)\) and \((1, -1, 2)\). We will then determine the intercepts on the coordinate axes and calculate \(a + b + c\).

### Step 1: Find the direction vectors of the line
The direction vector of the line joining points \(B(1, 1, 1)\) and \(C(1, -1, 2)\) can be calculated as:
\[ \vec{BC} = \vec{C} - \vec{B} = (1 - 1, -1 - 1, 2 - 1) = (0, -2, 1) \]

### Step 2: Find the direction vector from point A to point B
The direction vector from point \(A(-2, -2, 2)\) to point \(B(1, 1, 1)\) is:
\[ \vec{AB} = \vec{B} - \vec{A} = (1 - (-2), 1 - (-2), 1 - 2) = (3, 3, -1) \]

### Step 3: Find the normal vector to the plane
The normal vector \(\vec{n}\) to the plane can be found by taking the cross product of the two direction vectors \(\vec{AB}\) and \(\vec{BC}\):
\[ \vec{n} = \vec{AB} \times \vec{BC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 3 & -1 \\ 0 & -2 & 1 \end{vmatrix} \]
Calculating the determinant:
\[ \vec{n} = \hat{i}(3 \cdot 1 - (-1)(-2)) - \hat{j}(3 \cdot 1 - (-1)(0)) + \hat{k}(3 \cdot (-2) - 3 \cdot 0) \]
\[ = \hat{i}(3 - 2) - \hat{j}(3 - 0) + \hat{k}(-6) \]
\[ = \hat{i}(1) - \hat{j}(3) - \hat{k}(6) \]
Thus, the normal vector is:
\[ \vec{n} = (1, -3, -6) \]

### Step 4: Find the equation of the plane
The equation of the plane can be expressed as:
\[ 1(x + 2) - 3(y + 2) - 6(z - 2) = 0 \]
Expanding this:
\[ x + 2 - 3y - 6 - 6z + 12 = 0 \]
\[ x - 3y - 6z + 8 = 0 \]

### Step 5: Write the equation in intercept form
To find the intercepts, we rewrite the equation in intercept form:
\[ \frac{x}{-8} + \frac{y}{\frac{8}{3}} + \frac{z}{\frac{4}{3}} = 1 \]
From this, we can identify the intercepts:
- \(a = -8\)
- \(b = \frac{8}{3}\)
- \(c = \frac{4}{3}\)

### Step 6: Calculate \(a + b + c\)
Now, we calculate:
\[ a + b + c = -8 + \frac{8}{3} + \frac{4}{3} \]
Finding a common denominator (which is 3):
\[ = -\frac{24}{3} + \frac{8}{3} + \frac{4}{3} = -\frac{24 - 8 - 4}{3} = -\frac{20}{3} \]

### Final Answer
Thus, the value of \(a + b + c\) is:
\[ \boxed{-\frac{20}{3}} \]