What is the nature of the intersection of the set of planes `x+a y+(b+c)z+d=0,x+b y+(c+a)z+d=0a n dx+c y+(a+b)z+d=0?` (a). they meet at a point (b). they form a triangular prism (c). they pass through a line (d). they are at equal distance from the origin

Options

A. They meet at a point
B. They form a triangular prism
C. They pass through a line
D. They are at equal distance from the origin

Explanation

To determine the nature of the intersection of the given set of planes, we will analyze the equations of the planes step by step.

### Given Planes:
1. \( x + a y + (b+c) z + d = 0 \)
2. \( x + b y + (c+a) z + d = 0 \)
3. \( x + c y + (a+b) z + d = 0 \)

### Step 1: Write the equations in matrix form
We can express the system of equations in matrix form as follows:

\[ \begin{bmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -d \\ -d \\ -d \end{bmatrix} \]

### Step 2: Form the augmented matrix
The augmented matrix for the system of equations is:

\[ \begin{bmatrix} 1 & a & b+c & | & -d \\ 1 & b & c+a & | & -d \\ 1 & c & a+b & | & -d \end{bmatrix} \]

### Step 3: Analyze the determinant
To find the nature of the intersection, we need to analyze the determinant of the coefficient matrix:

\[ \begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{vmatrix} \]

Calculating this determinant, we can simplify it by using row operations or directly calculating it.

### Step 4: Calculate the determinant
Using the property of determinants, we can subtract the first row from the second and third rows:

\[ \begin{vmatrix} 1 & a & b+c \\ 0 & b-a & (c+a)-(b+c) \\ 0 & c-a & (a+b)-(b+c) \end{vmatrix} \]

This simplifies to:

\[ \begin{vmatrix} 1 & a & b+c \\ 0 & b-a & a-c \\ 0 & c-a & a-b \end{vmatrix} \]

The determinant of this matrix can be calculated, and if we find that it equals zero, it indicates that the planes do not intersect at a single point.

### Step 5: Conclusion about the intersection
Since the determinant is zero, it implies that the planes are either parallel or they intersect along a line. Given the structure of the equations, we conclude that the planes intersect along a line.

### Final Answer:
The correct option is (c) they pass through a line.