Let coefficient of `x^4` and `x^2` in the expansion of `(x+sqrt(x^2-1))^6+(x-sqrt(x^2-1))^6` is `alpha` and `beta` then `alpha-beta` is equal to

Options

A. `alpha - beta =-132`
B. `alpha + beta =60`
C. `alpha - beta =60`
D. `alpha+ beta =-30`

Answer

To solve the problem, we need to find the coefficients of \(x^4\) and \(x^2\) in the expansion of the expression \((x + \sqrt{x^2 - 1})^6 + (x - \sqrt{x^2 - 1})^6\).

### Step 1: Expand the expressions using the Binomial Theorem

Using the Binomial Theorem, we can expand both expressions separately.

1. **For \((x + \sqrt{x^2 - 1})^6\)**:
\[ (x + \sqrt{x^2 - 1})^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (\sqrt{x^2 - 1})^k \]
This gives us:
\[ = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (x^2 - 1)^{k/2} \]

2. **For \((x - \sqrt{x^2 - 1})^6\)**:
\[ (x - \sqrt{x^2 - 1})^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-\sqrt{x^2 - 1})^k \]
This gives us:
\[ = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-1)^k (x^2 - 1)^{k/2} \]

### Step 2: Combine the expansions

When we add both expansions, the odd powers of \(\sqrt{x^2 - 1}\) will cancel out:
\[ (x + \sqrt{x^2 - 1})^6 + (x - \sqrt{x^2 - 1})^6 = 2 \sum_{k \text{ even}} \binom{6}{k} x^{6-k} (x^2 - 1)^{k/2} \]

### Step 3: Identify coefficients for \(x^4\) and \(x^2\)

Now we need to find the coefficients of \(x^4\) and \(x^2\) from the combined expansion.

1. **Finding \(\alpha\) (coefficient of \(x^4\))**:
- The terms contributing to \(x^4\) can be derived from \(k=2\) and \(k=4\):
- For \(k=2\):
\[ \binom{6}{2} x^{6-2} (x^2 - 1)^{1} = 15 x^4 (x^2 - 1) = 15 x^4 - 15 x^2 \]
Coefficient of \(x^4\) is \(15\).
- For \(k=4\):
\[ \binom{6}{4} x^{6-4} (x^2 - 1)^{2} = 15 x^2 (x^4 - 2x^2 + 1) = 15 x^2 (x^4 - 2x^2 + 1) \]
Coefficient of \(x^4\) is \(15\).

Thus, the total coefficient for \(x^4\) is:
\[ \alpha = 15 + 15 = 30 \]

2. **Finding \(\beta\) (coefficient of \(x^2\))**:
- The terms contributing to \(x^2\) can be derived from \(k=0\), \(k=2\), and \(k=4\):
- For \(k=0\):
\[ \binom{6}{0} x^{6} (x^2 - 1)^{0} = 1 \]
Coefficient of \(x^2\) is \(0\).
- For \(k=2\):
\[ -15 \]
Coefficient of \(x^2\) is \(-15\).
- For \(k=4\):
\[ 15 \]
Coefficient of \(x^2\) is \(15\).

Thus, the total coefficient for \(x^2\) is:
\[ \beta = 0 - 15 + 15 = 0 \]

### Step 4: Calculate \(\alpha - \beta\)

Finally, we calculate:
\[ \alpha - \beta = 30 - 0 = 30 \]

### Conclusion

Thus, the value of \(\alpha - \beta\) is \(30\).