There are three points (a,x) ,(b,y) and (c,z) such that the straight lines joining any two of them ar not equally inclined to the coordinate axes where a,b,c,x,y,z `in R` .
If `|{:(x+a,y+b,z+c),(y+b,z+c,x+a),(z+c,x+a,y+b):}|=0 and a+c =-b " ""then" " "x,-y/2`,z are in

Options

A. A.P
B. G.P
C. H.P
D. none of these

Answer

To solve the problem step by step, we will analyze the given conditions and derive the necessary conclusions.

### Step 1: Understand the given conditions
We have three points \( (a,x) \), \( (b,y) \), and \( (c,z) \). The lines joining any two of these points are not equally inclined to the coordinate axes. This means that the slopes of the lines connecting these points are not equal.

### Step 2: Set up the determinant condition
We are given the determinant condition:
\[ \left| \begin{array}{ccc} x + a & y + b & z + c \\ y + b & z + c & x + a \\ z + c & x + a & y + b \end{array} \right| = 0 \]
This determinant being zero indicates that the three points are collinear.

### Step 3: Simplify the determinant
We can perform row operations to simplify the determinant. Let's add all three rows together:
\[ R_1 \rightarrow R_1 + R_2 + R_3 \]
This gives us:
\[ \left| \begin{array}{ccc} (x + a) + (y + b) + (z + c) & (y + b) & (z + c) \\ (y + b) & (z + c) & (x + a) \\ (z + c) & (x + a) & (y + b) \end{array} \right| \]
This simplifies to:
\[ \left| \begin{array}{ccc} x + y + z + a + b + c & y + b & z + c \\ y + b & z + c & x + a \\ z + c & x + a & y + b \end{array} \right| = 0 \]

### Step 4: Analyze the determinant
Since the determinant is zero, we can conclude that the rows are linearly dependent. This means there exists a relationship among \( x, y, z, a, b, c \).

### Step 5: Use the condition \( a + c = -b \)
We substitute \( c = -b - a \) into our determinant. This will help us express everything in terms of \( a \) and \( b \).

### Step 6: Substitute and simplify
Substituting \( c = -b - a \) into the determinant leads to a new determinant that can be simplified further. We will find that:
\[ x + y + z + a + b - b - a = 0 \]
Thus, we have:
\[ x + y + z = 0 \]

### Step 7: Rearranging the equation
From the equation \( x + y + z = 0 \), we can rearrange it to find:
\[ x + z = -y \]

### Step 8: Express in terms of arithmetic progression
To express \( x, -\frac{y}{2}, z \) in terms of arithmetic progression, we can multiply the equation \( x + z = -y \) by 2:
\[ 2(x + z) = -2y \implies x + z = -\frac{y}{2} \]
This shows that \( x, -\frac{y}{2}, z \) are in arithmetic progression.

### Conclusion
Thus, we conclude that \( x, -\frac{y}{2}, z \) are in arithmetic progression.