Given,

$\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$

$\Rightarrow \frac{x^2}{k-\frac{5}{2}}-\frac{y^2}{k-\frac{7}{3}}=1$

On comparing with equation of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

We get, $a^2=k-\frac{5}{2} \& b^2=k-\frac{7}{3}$

Now to define $a^2$the value of $k\in \left(\frac{5}{2},\infty \right)$ ,

And to define $b^2$ the value of $k\in \left(\frac{7}{3},\infty \right)$

Now on taling intersection on value of $k$ we get, $k\in \left(\frac{5}{2},\infty \right)$