Suppose `k in R`. Let a be the coefficient of the middle term in the expansion of `(k/x+x/k)^(10)` and b be the term independent of x in the expansion of `((k^(2))/x+x/k)^(10)`. If `a/b=1`, then k is equal to

Options

A. 1
B. 2
C. `-3`
D. any non-zero number

Answer

To solve the problem, we need to find the value of \( k \) given the conditions regarding the coefficients \( a \) and \( b \) from the expansions of the expressions. Let's break it down step by step.

### Step 1: Identify the middle term in the expansion of \( \left(\frac{k}{x} + \frac{x}{k}\right)^{10} \)

The number of terms in the expansion of \( (x + y)^n \) is \( n + 1 \). Here, \( n = 10 \), so the number of terms is \( 10 + 1 = 11 \). The middle term is the \( 6^{th} \) term (since there are 5 terms on either side).

The general term in the binomial expansion is given by:
\[ T_{r+1} = \binom{n}{r} x^{n-r} y^r \]

For our expression, we have:
\[ T_{r+1} = \binom{10}{r} \left(\frac{k}{x}\right)^{10-r} \left(\frac{x}{k}\right)^r \]

To find the \( 6^{th} \) term, we set \( r = 5 \):
\[ T_6 = \binom{10}{5} \left(\frac{k}{x}\right)^{5} \left(\frac{x}{k}\right)^{5} \]
\[ = \binom{10}{5} \frac{k^5}{x^5} \cdot \frac{x^5}{k^5} = \binom{10}{5} \]

Thus, the coefficient \( a \) of the middle term is:
\[ a = \binom{10}{5} \]

### Step 2: Find the term independent of \( x \) in the expansion of \( \left(\frac{k^2}{x} + \frac{x}{k}\right)^{10} \)

Again, we need to find the \( 6^{th} \) term in this expansion. The general term is:
\[ T_{r+1} = \binom{10}{r} \left(\frac{k^2}{x}\right)^{10-r} \left(\frac{x}{k}\right)^r \]

Setting \( r = 5 \) for the \( 6^{th} \) term:
\[ T_6 = \binom{10}{5} \left(\frac{k^2}{x}\right)^{5} \left(\frac{x}{k}\right)^{5} \]
\[ = \binom{10}{5} \frac{(k^2)^5}{x^5} \cdot \frac{x^5}{k^5} = \binom{10}{5} \frac{k^{10}}{k^5} = \binom{10}{5} k^5 \]

Thus, the term independent of \( x \) is:
\[ b = \binom{10}{5} k^5 \]

### Step 3: Set up the equation \( \frac{a}{b} = 1 \)

Given that \( \frac{a}{b} = 1 \):
\[ \frac{\binom{10}{5}}{\binom{10}{5} k^5} = 1 \]

This simplifies to:
\[ \frac{1}{k^5} = 1 \implies k^5 = 1 \]

### Step 4: Solve for \( k \)

Since \( k^5 = 1 \), the real solution is:
\[ k = 1 \]

### Conclusion

The value of \( k \) is:
\[ \boxed{1} \]