Binomial Theorem - Coefficients in Products of Binomial Expansions

In the binomial expansion of \({{\left( x+\frac{k}{x} \right)}^{7}}\), where \(k\) is a positive constant, the coefficients of \({{x}^{3}}\) and \({{x}^{5}}\) are the same. Find the value of \(k\).

Using the value of \(k\) found in part (a), find the coefficient of \({{x}^{5}}\) in the expansion of \((1-3{{x}^{2}}){{\left( x+\frac{k}{x} \right)}^{7}}\).