The coefficient of `x^(6) a^(-2)` in the expansion of `((x^(2))/(a)-(a)/(x))^(12)`, is

Options

A. `""(12)C_(6)`
B. `-""^(12)C_(5)`
C. 0
D. none of these

Answer

To find the coefficient of \( x^6 a^{-2} \) in the expansion of \( \left( \frac{x^2}{a} - \frac{a}{x} \right)^{12} \), we can follow these steps:

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the binomial expansion of \( (A + B)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} A^{n-r} B^r \]
Here, \( A = \frac{x^2}{a} \), \( B = -\frac{a}{x} \), and \( n = 12 \).

### Step 2: Write the General Term for the Given Expansion
For our case:
\[ T_{r+1} = \binom{12}{r} \left( \frac{x^2}{a} \right)^{12-r} \left( -\frac{a}{x} \right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{12}{r} \left( \frac{x^{2(12-r)}}{a^{12-r}} \right) \left( -\frac{a^r}{x^r} \right) \]
Combining the terms:
\[ T_{r+1} = \binom{12}{r} (-1)^r \frac{x^{24 - 3r}}{a^{12 - 2r}} \]

### Step 3: Set Up the Conditions for Coefficients
We need the term where the power of \( x \) is 6 and the power of \( a \) is -2:
1. From \( x^{24 - 3r} = x^6 \):
\[ 24 - 3r = 6 \implies 3r = 18 \implies r = 6 \]

2. From \( a^{-(12 - 2r)} = a^{-2} \):
\[ -(12 - 2r) = -2 \implies 12 - 2r = 2 \implies 2r = 10 \implies r = 5 \]

### Step 4: Analyze the Results
We found two different values for \( r \):
- From \( x \) term: \( r = 6 \)
- From \( a \) term: \( r = 5 \)

Since there is no single value of \( r \) that satisfies both conditions, it means that the term \( x^6 a^{-2} \) does not exist in the expansion.

### Step 5: Conclusion
Thus, the coefficient of \( x^6 a^{-2} \) in the expansion is:
\[ \text{Coefficient} = 0 \]