a and b are non-collinear vectors. If `c=(x-2) a+b and d=(2x+1)a-b` are collinear vectors, then find the value of x.

Answer

To solve the problem step by step, we will analyze the vectors \( c \) and \( d \) and use the property of collinearity.

### Step 1: Write down the vectors
We have:
\[ c = (x - 2) a + b \]
\[ d = (2x + 1) a - b \]

### Step 2: Set up the collinearity condition
Since \( c \) and \( d \) are collinear, there exists a scalar \( \lambda \) such that:
\[ c = \lambda d \]
Substituting the expressions for \( c \) and \( d \):
\[ (x - 2) a + b = \lambda \left( (2x + 1) a - b \right) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives:
\[ (x - 2) a + b = \lambda (2x + 1) a - \lambda b \]

### Step 4: Rearrange the equation
Rearranging the equation, we can group the terms involving \( a \) and \( b \):
\[ (x - 2) a + b + \lambda b = \lambda (2x + 1) a \]
This simplifies to:
\[ (x - 2) a + (1 + \lambda) b = \lambda (2x + 1) a \]

### Step 5: Equate coefficients of \( a \) and \( b \)
Now, we can equate the coefficients of \( a \) and \( b \):
1. Coefficient of \( a \):
\[ x - 2 = \lambda (2x + 1) \]
2. Coefficient of \( b \):
\[ 1 + \lambda = 0 \quad \Rightarrow \quad \lambda = -1 \]

### Step 6: Substitute \( \lambda \) into the first equation
Substituting \( \lambda = -1 \) into the equation for the coefficient of \( a \):
\[ x - 2 = -1(2x + 1) \]
This simplifies to:
\[ x - 2 = -2x - 1 \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ x + 2x = -1 + 2 \]
\[ 3x = 1 \]
\[ x = \frac{1}{3} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{1}{3}} \]